My current tzatziki sauce recipe is yogurt, garlic, cucumbers, and a little lemon or lime juice - so it seems like you could do that without the cucumber and it'd still have some goodness. (I also often throw in minced dried onions, but that's me, and not especially traditional.)

If it were me, I think I'd mix it with garlic, a few suitable herbs if handy (dill? tarragon? parsley?) and a bit of the lime juice. I'd put a bit of the chicken drippings in with the rice, stir a bit, and then stir the yogurt in. The same thing over the new potatoes (roasted?) would also be yummy to me: I really like the contrast between the creamy yogurt and the either the toothiness of the rice, or the crispness of roasted veggies. Or you could put it over the broccoli. Or some combination of those three.