Line a colander with two layers of cheesecloth and place it over a medium bowl. Place the yogurt on the cheesecloth and cover the colander with plastic wrap. Allow yogurt to drain overnight. (If you buy Greek Style Yogurt, you will not need to drain.)
Lay grated cucumber on a plate lined with paper towel; allow to drain 1 to 2 hours.
Combine the drained yogurt, cucumber, garlic, parsley, mint, lemon juice, salt, and pepper in a bowl. Refrigerate for at least 2 hours before serving.