Line a sieve with cheesecloth (or a doubled paper towel), and place over a bowl.
Spoon the yogurt into the sieve, cover and refrigerates for at least 4-6 hours, or as long as overnight.
Place the grated cucumber in a sieve set over a bowl, sprinkle with salt and let drain for 30 mins.
Rinse off salt and gently squeeze the cucumber dry.
In a bowl, combine the drained yogurt, garlic, olive oil, cucumber, lemon juice, and chopped mint. Season to taste with ground pepper.
Cover and refrigerate until serving. Mixture may look as thought it is separating (depending on how long it is refrigerated); simply stir and serve.
Garnish with fresh mint sprigs, if desired.