I don't think it works to eliminate one thing. It's very uncommon to have only one food sensitivity, if you have them. Elimination diets are the gold standard, but how do you do one?
This is a hard question. I went on a paleo diet, and after a few weeks also went off eggs, and I got tremendous relief. But what if I was one of the people allergic to, eg, beef? You have to make up your own way to do it, according to your best guesses + what you can tolerate doing.
Another approach would be to do some allergy testing, and then remove all of the 'bad' foods and see what happens. Just yesterday a friend recommended Bio Tek's testing as being fully accurate for her. But that has to be done through a doctor, and I have no idea about cost.