In a large mixing bowl, combine the Greek yogurt, grated cucumber, dill, lemon juice, minced garlic, salt, and black pepper. Mix well until all ingredients are thoroughly combined.
Add the shredded chicken, red onion, and cherry tomatoes to the yogurt mixture. Gently fold the ingredients together until the chicken and vegetables are evenly coated with the tzatziki dressing.
Serve the tzatziki chicken salad over a bed of mixed greens or lettuce.
For best flavor, chill the salad in the refrigerator for at least 30 minutes before serving to allow the flavors to meld.