Do pets need variety in their diets?

Traditionally, veterinarians, breeders and other pet-related experts have preached that a pet's diet should remain consistent; if you do decide to change what you're feeding your dog or cat, the direction has always been to do so very slowly and gradually. And certainly, petfood companies want to build brand loyalty and follow this line of thinking.
But lately, we're hearing from people who say, "Nonsense. If humans can eat a variety of foods, why not pets? And would you want to eat the same exact food meal after meal, day after day?" (See also this item and this column.)
In fact, some petfood companies, such as The Honest Kitchen and Nature's Variety, have developed their lines based on this theory.
Is this just a case of anthropomorphism or a countering marketing strategy to traditional petfood promotion? Is there scientific evidence that feeding a variety of foods offers nutritional benefits to pets? For that matter, does research back the conventional wisdom that a dog or cat should stick to one food exclusively?
If you know of studies backing either theory (or something else entirely), please share!