In a mixing bowl, combine olive oil, lemon juice, garlic, oregano, salt, and pepper. Add chicken pieces and toss to coat. Cover and marinate in the fridge for at least 30 minutes or up to 4 hours.
While the chicken marinates, prepare the tzatziki sauce by combining Greek yogurt, grated cucumber, lemon juice, garlic, olive oil, dill, and salt in a small bowl. Mix well, cover, and chill until ready to use.
Heat a skillet or grill pan over medium-high heat. Add the marinated chicken pieces and cook for 4–5 minutes per side, or until golden and cooked through (internal temperature should reach 165°F).
Warm the pita breads and layer with lettuce, cherry tomatoes, sliced red onion, and grilled chicken. Spoon tzatziki sauce over the top and add feta or olives if desired.
Fold the pita and serve immediately.