Place a colander in a large bowl. Line colander with 4 layers of cheesecloth, allowing cheesecloth to extend over outside edges. Spoon yogurt into colander. Cover loosely with plastic wrap, and refrigerate 12 hours. Spoon drained yogurt into a bowl, and discard liquid. Cover and refrigerate drained yogurt.
Place cucumber in colander; sprinkle with 1/2 teaspoon salt. Toss well to combine. Drain 15 minutes. Place cucumber on paper towels; squeeze until barely moist. Combine drained yogurt, cucumber, remaining 1/4 teaspoon salt, mint, pepper, and garlic. Drizzle with oil.