Another favorite classic Italian dish at my house is Risotto. This is a rice dish made with Arborio (or more simply, medium grain rice). It is a more time-consuming recipe than most rice dishes. Instead of just dumping the water and rice into a pot and forgetting about it for 20 minutes, you gradually add broth as it's absorbed by the rice. Even though you do have to watch it for 20 min., it is very easy to make and the you end up with a nice creamy dish.
Like Frittata, this dish is versatile and you can easily substitute ingredients, but you always use Parmesan cheese.
Here is a very basic recipe for risotto.
Here is a recipe for Risotto Primavera. (Personally, I leave out the icky peas.)
If all you have in your pantry is long grain rice, it'll still work but the medium or short grain is better. I've never tried it with brown rice.
UPDATE: I tried it with brown rice. DON'T do it. It was a total failure. The brown rice takes too long to absorb the liquid so it evaporates before it cooks the rice.