Starches are omnipresent in many foods, which makes them difficult to avoid. Starches make up a large part of the carbohydrates consumed by Americans. Foods made from starchy vegetables, grains or their flours, such as french fries, baked potatoes, breads, pasta, rice, cookies and cakes, are all high in starches.f you are not sure whether a food contains starches or not, look at the ingredient list. Anything made from wheat or other grains and their flour contains starches. Potatoes and other starchy vegetables are other starch-rich foods that you should get rid of to help you stick to your no-starch diet.