If you like raw carrots, you'll probably enjoy raw beets. They're similarly crunchy, and quite tasty. Try them in salads, either cubed or shredded, or cut them into matchsticks to eat as a healthy snack.
While most beet recipes require cooked beets in one form or another, such as steamed or roasted beets, I have seen several recipes for raw beet salads. I'd approach beets with caution, though, if your body isn't used to them, because they can have a laxative effect.
If you use them raw, remember that they can be tough, so cut them small, salt them, and leave them in a colander to drain for 10 minutes. Usually, though, they're better cooked.