Preheat a grill to medium-high heat (375 to 450 degrees Fahrenheit).
Wash and peel the beets with a vegetable peeler, then dice them (take care: the beet juice can stain!).
In a bowl, toss the beets with the olive oil, Italian seasoning, salt, and plenty of fresh ground pepper.
Place the beets on a large sheet of aluminum foil and fold up the sides. Place a second piece of foil on top and seal the seams. Poke several holes in the packet with fork (to allow stem to escape).
Place the packet directly on the grill grates and cook for 35 to 50 minutes until tender (open and test with a fork to assess doneness; exact timing depends on the size of pieces and exact grill temperature). Taste and add additional salt as necessary. To serve, drizzle with olive oil and crumble on feta cheese (optional).