Tzatziki sauce: combine yogurt, garlic, dill, and lemon juice in a bowl. Stir in diced cucumber and season to taste with salt and pepper. Refrigerate.
Chicken Marinade: Add chicken to a large bowl or ziplock bag. Combine marinade ingredients and pour over chicken. Stir well to coat and marinate for 1-4 hours.
Grill: Heat grill to medium and oil the grates. Remove chicken from marinade, letting any excess sauce drip off. Grill chicken for a few minutes on each side, flipping once, until cooked through. Remove to a plate to rest before chopping into bite-size pieces.
Assemble Gyros: Toast or warm your pitas. Layer with a big scoop of chopped chicken, lettuce, tomato, red onion, cucumber and tzatziki.
