When we lived in Cyprus, we used to get a Greek beetroot salad, there are two versions, one has greek yogurt added, that one is yummy with grilled food, very refreshing on a warm day
1. Boil beetroots and cut them into cubes.
2. In a bowl mix the garlic, the yogurt, the mayonnaise, the olive oil and the salt.
3. Add the beetroots cubes and stir gently.
4. Sprinkle with fresh pepper on top.
If you have some nice fresh beetroot why not just wrap in foil, roast on the bbq for a while (1/2 hour say?), peel off the skin and eat with a generous splash of vinegar. Simple yet effective. Beetroot cooked simply like this is great with things like bubble and squeak or fish cakes and I'm sure it would be nice with bbq meats and salads.
(another thing that is simple to do this way is corn on the cob. Just wrap in foil with butter and lots of black pepper)