Pour yogurt into a strainer set over a bowl. Cover with plastic wrap, refrigerate and let drain until thickened, about 2 hours. Discard liquid in bowl, spoon yogurt into another bowl and stir in cucumber, mint, garlic, and salt and pepper to taste.
Preheat a gas grill on high, covered, for 10 minutes, then turn heat to medium. Oil a grill rack. Sprinkle salmon fillets with salt and pepper. Place salmon fillets, flesh side down, on oiled rack and cook, covered, for 4 minutes. Turn salmon carefully and continue to grill until just cooked through, 4 to 6 minutes longer. Serve with yogurt sauce on the side.