The adage that you are what you eat doesn't quite ring true for some people. A growing group is swearing by the idea: You are what you don't eat.

Elimination diets—which call for cutting out a certain food or group of foods—are having a moment. People often start the diets in an effort to cure a symptom that doesn't necessarily seem worth seeing a doctor about: headaches, skin irritation, joint pain, digestive problem or just feeling tired. Maybe cutting out certain foods will help, they think. Weight loss often isn't the...