Make the Tzatziki: In a medium bowl, combine Greek yogurt, grated cucumber (squeeze out excess moisture), minced garlic, lemon juice, dill, salt, and pepper. Stir well and refrigerate for at least 30 minutes to allow flavors to meld.
Marinate the Chicken: In a large bowl, combine chicken pieces, olive oil, minced garlic, lemon juice, oregano, thyme, salt, and pepper. Mix well and marinate for at least 30 minutes (or up to several hours) in the refrigerator.
Cook the Chicken: Heat a large skillet or grill pan over medium-high heat. Add the marinated chicken and cook for 6-8 minutes, or until cooked through and lightly browned, stirring occasionally.
Warm the Pitas: Warm pita breads in a dry skillet, microwave, or toaster until pliable.
Assemble the Gyros: Spread tzatziki sauce on each pita bread. Top with cooked chicken, sliced red onion, and sliced tomato. Add feta cheese and Kalamata olives if desired.
Serve immediately and enjoy!