Brush the beats with olive oil and place over medium-low coals. Cook raw beats slowly over heat until they are easily pierced, 1 to 1 1/2 hours.
Remove from the barbecue, peel, then cut into thick slices.
Meanwhile, blend the dressing ingredients. Combine garlic, onion, tomato, cilantro, cumin, curry powder, vinegar and olive oil. Toss with the beets. Taste for seasoning.
Serve warm or at room temperature.
Note: Canned beets can be substituted. Either skip first two steps, or place beets on the grill for only about 5 minutes to pick up some of the smoky goodness. Sliced beets should probably not be put on the grill unless you line it with some aluminum foil.