Prepare the chicken by shredding it into bite-sized pieces. If cooking from scratch, boil or grill the chicken breasts until fully cooked, about 15-20 minutes. Let cool before shredding.
Make the tzatziki sauce by combining Greek yogurt, grated cucumber, minced garlic, dill, lemon juice, and olive oil in a bowl. Mix well and squeeze out excess moisture from the cucumber.
Combine the tzatziki sauce with the shredded chicken in a mixing bowl. Gently fold in halved cherry tomatoes and chopped red onion. If using, sprinkle feta cheese on top and mix gently.
Chill the salad for at least 30 minutes to allow flavors to meld. Serve on a bed of mixed greens or lettuce.