The answer to this is really, anything. After all, the cuisine of Italy is legendary, and the foods of the Umbrian area are worthy of this same reputation.
Because Umbria is so robust and such an ideal place for agriculture, its foods and wines are world famous. A day in Montefalco will introduce you to some of the best red wines in the world, while the Orvieto area would allow you to experience a delectable white or two.
What else will the savvy traveler eat while visiting the region? Frequently summarized as "rustic", the foods of Umbria are more seasonal than anything else. Though the most famous dishes do use only a few ingredients, they will be the finest and freshest you have ever enjoyed. The diet relies on grains, produce, excellent olive oil, and herb, though meat plays a major role. Pastas and risottos are a frequent delight; soups are common and include lentils, fava beans, and nuts that are part of the standard diet.
Do a bit of research ahead of your visit to discover which foods are in season during a visit. For example, if you learn you are arriving during the height of asparagus season, make a pointed effort to indulge in as many dishes crafted with it as possible as you will rarely encounter anything that will compare.
The following is a brief list of the foods or dishes that are considered specialties of the area:
Cheese - Goat's milk and pecorino (sheep's) milk cheeses are the pride of Umbria. Ripened and mature, or fresh and soft, they are some of the most commonly found in the region.
Mushrooms - Truffles are actually common here and appear frequently in the dishes. Indulge in them as much as you can as they are used in surprisingly delicious and unusual ways.
Meat - Cured meats, such as salami and sausages, are common as part of an antipasti or simple meal. The diet in general is quite heavy where meat is concerned, with pork as a primary choice, though lamb and game are common as well. Seafood is not as popular here as in other parts of Italy due to the lack of coast.
Bread - Umbrian breads are without salt, tend to be flat, and frequently fried in a pan. They will often be stuffed, and it is torta al testo that is the most popular bread throughout the Umbrian region.