In a large mixing bowl, combine the Greek yogurt, grated cucumber, minced garlic, dill, lemon juice, olive oil, salt, and pepper. Mix until well blended.
Add the shredded chicken, red onion, and cherry tomatoes to the bowl. Gently fold the ingredients together until everything is evenly coated with the tzatziki mixture.
If using, sprinkle the crumbled feta cheese on top and gently mix it in.
Taste and adjust seasoning with more salt, pepper, or lemon juice if desired.
Cover the bowl and refrigerate for at least 30 minutes to allow the flavors to meld. Serve chilled.