How to Do an Elimination Diet
In the Elimination Diet you remove suspected allergenic foods from your diet. Food sensitivity and allergy reactions can include upset stomach, diarrhea, headache, brain fog (you can't think straight), physical illness, arthritis, ADD/ADHD, autoimmune disorders, asthma, allergies and other symptoms. Surprisingly, mood disorders and feelings of anger, sadness or stress can also be the result of eating the wrong kinds or amounts of food.
Foods are removed for 3-4 weeks and gradually reintroduced to see what happens. You can go whole hog and remove a lot of foods or you can go minimalist and remove just the one(s) that you suspect are the culprits.
Deluxe Elimination Diet
To go for the Deluxe Elimination Diet remove gluten, dairy, soy, eggs, corn, pork, beef, chicken, beans/lentils, coffee, citrus fruits, nuts, and nightshade vegetables (tomatoes, peppers, potatoes) from your diet. Basically, you can eat turkey, fish, lamb, rice, and most fruits and vegetables. These acceptable foods were chosen because they are not normally allergenic. 
Picture: There is even hypoallergenic pet food for dogs and cats.
For inspiration, think about the expensive bags of 'hypoallergenic' Fido Chow made of lamb and rice or salmon and rice at the pet store. The Deluxe Elimination Diet is a difficult diet but may be worth it if you think your health problems are due to food allergies or intolerances. Sadly, they don't make bags of specially formulated 'people chow' to try instead of formulating your own diet.
Lite Elimination Diet
People who don't want to revamp their entire diet can try the Lite Elimination Diet. Basically you just eliminate the individual foods that you suspect are giving you grief (or in this case, intestinal distress). Make sure to read labels; it is surprising where you find different foods!
Evaluate Your Symptoms
In either case, after 4-5 weeks of the diet monitor how you feel and what symptoms are occurring:
I feel the same: If you feel the same as when you started the diet, the foods eliminated are unlikely to have been the problem. In this case, you can look at what you are still eating or see if your problems are caused by something other than food. Sometimes stress can cause gastrointestinal problems. Also: Reread all the ingredients in the food you have been eating. You may find that you are still eating a suspected food. I feel better: If you feel better or your symptoms are reduce or eliminated, you may have a food allergy or sensitivity to one or more of the foods you eliminated from your diet. To determine the culprit, begin to gradually introduce foods and see how you feel. Add one food and monitor your symptoms for two days. If you have no noticeable symptoms, reintroduce another food for one day and monitor your symptoms.
Symptoms to watch out for include: fatigue, intestinal upsets, stomach or bowel change, bloating or water retention, headache, joint pain, inflammation, mood changes such as anger or sadness, brain fog, asthma, sinus problems, and any other negative physical or mental problems.
One of my friends discovered she was allergic to egg. It took her over a year to find out that egg intake caused her have edema all over. It is amazing how many foods use egg (even chips) so it was hard to eliminate.
Susan's Food Intolerance Tale:
I did a version of the Lite Elimination Diet because I suspected I was having a negative reaction to carbohydrates. I cut out refined carbohydrates, including grains, and refined sugars from my diet for around a month. I continued to eat most other whole natural foods. After a few days I felt wonderful, lost a few pounds, and had boundless energy and glowing skin. I could tell something was happening but I attributed it to reducing carbohydrates. However, I was wrong (which shows you that sometimes even nutritional professionals can suspect the wrong foods).
At the end of the month I ate a handful of wheat thins. I have never felt so sick as I did a few hours later! It turned out I was gluten sensitive not carbohydrate sensitive, which I confirmed with a little more testing on my part. I now eat wheat and gluten free. 
Picture: Pink rhododendron blossoms.
My friends wondered if my strict adherence to gluten free was worth forgoing donuts. It certainly is worth it! These are the benefits I get from continuing to avoid treats made from wheat.
I hardly ever get headaches (maybe 2-3 per year). I used to get frequent headaches. My brain fog has disappeared totally. I used to get brain fog after I ate and be unable to think straight. I hated it because I don't like my head feeling fuzzy. Now I never get brain fog except for a few times when I accidently have consumed wheat. I sleep better and fall asleep quickly. Now I am either wide awake or fast asleep. I don't lie around in a half-awake stupor. My digestive system is much happier and runs smoothly. I have lots of energy. I feel really happy almost all the time. I have always been a positive person but eliminating gluten has made me almost disgustingly happy. This uptick in mood has been noticed by other family members and friends as well.
If I accidently eat wheat or gluten I get a headache, and become grumpy and angry (over nothing) fairly quickly. I suspect other gastrointestinal problems would develop if I continued to eat wheat but quite frankly, it is not worth it.
If I had been more observant I might have noticed my wheat intolerance earlier. Even as a child, I didn't like pasta. When I ate large amounts of pasta or bread I usually got brain fog and fell into a sort of sleepy stupor. I thought it was due to eating too much sugar or sleeping too little. As I got older I developed many other symptoms of celiac disease; I just didn't recognize them.
Digestion & Terminology
Allergy: a very specific immune response when the food is eaten (or in some causes touched or inhaled). In allergies either the immunoglobulin E (IgE) or T-cells over react to the food. IgE releases histamine and other chemicals which can cause hives, itching, runny nose, watery eyes, wheezing, feeling sick or anxious and anaphylaxis. The offending food triggers an immediate response. People with allergies should avoid the food since allergies can escalate rapidly. Food allergies can also develop and change over time so you can become either more or less sensitive to a food.
Sensitivity: People have an unpleasant reaction; such as acid reflux, stomach cramps, fatigue or mood changes; to certain foods. The causes of this may be unknown.
Some people believe that food sensitivities are provoked by a immunoglobulin G (IgG) immune response. Please do not be fooled by people who advocate blood testing for serum IgG as an indicator of food sensitivities. Science doesn't support this belief and you may be eliminating innocent and tasty foods (Hon et al. 2014). IgG concentrations measure a person's exposure to a substance; they are not an indication of an allergenic response. The exception is a diagnostic test for celiac disease.*
Numerous studies show that as serum concentrations of IgG for a food rise, tolerance for that food increases. Children who recovered at an early age from cow milk allergies had higher concentrations of serum IgG when compared to children whose allergies persisted as they got older (Savilahti et al. 2010, Savilahti and Savilahti 2013). Likewise, two-year old toddlers with egg and milk allergies were more likely to be able to consume milk and eggs at age four if they had high concentrations of serum IgG (Tomicić et al. 2009). Similar work showed that children desensitized to peanuts had higher concentrations of IgG (Jones et al. 2009, Vickery et al. 2013). Low concentrations of IgG may actually signify a food allergy. A recent study reported that children with milk and egg allergies had low concentrations of IgG (Savilahti et al. 2012).
Interesting Fact: Cow's milk allergy (CMA) affects 2-3% of infants. In most cases the allergy disappears as the child gets older (Savilahti and Savilahti 2013).
Intolerance: The body lacks a specific enzyme to digest the food or the enzyme is malformed or something else is happening to prevent the food from being taken up by the intestine (i.e. lactose intolerance, celiac disease). Intolerance is due to a metabolic and/or gastrointestinal reaction. There is a movement to change 'food intolerance' to 'food malabsorption' especially with fructose intolerance. In fact some people are quite vocal about it. It is true that malabsorption better describes what actually happens in the body. Includes:
Lactose Intolerance Celiac Disease (this is a non-IgE allergic reaction) Fructose Intolerance (Malabsorption)
Malabsorption: The body lacks a specific enzyme to digest the food, the enzyme is malformed or the body is otherwise unable to digest the food.
Pharmacological food intolerances: Pharmacologically active substances, such as biogenic amines, in foods that can cause a reaction. Offending foods include:
Phenylethylamine in chocolate Histamine in tomatoes, red wine and aged cheese Tyramine in chocolate and aged cheese Serotonin in bananas and nuts Glutamate (glutamate intolerance) Caffeine
*The exception to IgG tests are two procedures used to diagnose celiac disease:
An older test is called IgG-gliadin antibody test. A positive test result indicates an immunological reaction to gluten. It may or may not mean you have celiac disease (people with celiac disease also have intestinal damage and inflammation). People who test positive for this test usually do have a gluten sensitivity. If you test negative on this test you most likely do not have a gluten disorder. The newer test is called deamidated Gliadin Peptide antibody test which is used to help screen celiac disease patients by detecting gut damage. This is the test usually used in the United States today. If this test is positive you likely have celiac disease. If this test is negative, you probably do not have celiac disease BUT you may have gluten sensitivity.
Gliadin is one of the proteins found in gluten. So if you have been on a gluten free diet these tests will not be accurate.
References:
Jones SM, Pons L, Roberts JL, Scurlock AM, Perry TT, Kulis M, Shreffler WG, Steele P, Henry KA, Adair M, Francis JM, Durham S, Vickery BP, Zhong X, Burks AW. Clinical efficacy and immune regulation with peanut oral immunotherapy. J Allergy Clin Immunol. 2009;124:292-300. Pubmed. doi: 10.1016/j.jaci.2009.05.022 (full text) Hon KL, Poon TC, Pong NH, Wong YH, Leung SS, Chow CM, Leung TF. Specific IgG and IgA of common foods in Chinese children with eczema: Friend or foe. J Dermatolog Treat. 2014 Dec;25(6):462-6. Pubmed. doi: 10.3109/09546634.2013.848262 Savilahti EM, Savilahti E. Development of natural tolerance and induced desensitization in cow's milk allergy. Pediatr Allergy Immunol. 2013;24:114-21. Pubmed. doi: 10.1111/pai.12004 Savilahti EM, Rantanen V, Lin JS, Karinen S, Saarinen KM, Goldis M, Mäkelä MJ, Hautaniemi S, Savilahti E, Sampson HA. Early recovery from cow's milk allergy is associated with decreasing IgE and increasing IgG4 binding to cow's milk epitopes. J Allergy Clin Immunol. 2010;125:1315-1321. Pubmed. doi: 10.1016/j.jaci.2010.03.025 (full text) Savilahti EM, Viljanen M, Kuitunen M, Savilahti E. Cow's milk and ovalbumin-specific IgG and IgA in children with eczema: low β-lactoglobulin-specific IgG4 levels are associated with cow's milk allergy. Pediatr Allergy Immunol. 2012;23:590-6. Pubmed. doi: 10.1111/j.1399-3038.2012.01277.x Tomicić S, Norrman G, Fälth-Magnusson K, Jenmalm MC, Devenney I, Böttcher MF. High levels of IgG4 antibodies to foods during infancy are associated with tolerance to corresponding foods later in life. Pediatr Allergy Immunol. 2009;20:35-41. Pubmed. doi: 10.1111/j.1399-3038.2008.00738.x Vickery BP, Lin J, Kulis M, Fu Z, Steele PH, Jones SM, Scurlock AM, Gimenez G, Bardina L, Sampson HA, Burks AW. Peanut oral immunotherapy modifies IgE and IgG4 responses to major peanut allergens. J Allergy Clin Immunol. 2013;131:128-34. Pubmed. doi: 10.1016/j.jaci.2012.10.048 (full text)