Vegetables and Diabetes. Most vegetables contain fiber and are naturally low in fat and sodium (unless they are canned or frozen in sauces). Starchy vegetables, such as potatoes and corn, aren't included in this category. They are considered part of the breads, grains, and other starches group. Best Choices: 1  Fresh vegetables, eaten raw or lightly steamed, roasted, or grilled. 2  Plain frozen vegetables, lightly steamed. 3  Low sodium or unsalted canned vegetables.