Stir the olive oil, lemon juice, garlic, oregano, salt, and pepper together. Toss with the chicken and marinate for 15 to 30 minutes in the refrigerator while you prepare the sauce.
For the tzatziki, mix the Greek yogurt, squeezed cucumber, 1 tablespoon lemon juice, grated garlic, and dill. Chill until serving.
Heat a large skillet over medium-high. Lift the chicken from the marinade, letting excess drip away, and arrange it in a single layer; cook in batches if needed.
Cook for 4 to 6 minutes, turning once or twice, until browned and the thickest pieces reach 165°F (74°C). Rest the chicken for 3 minutes before serving.
Warm the pitas in a dry skillet or wrapped in foil in a 300°F (150°C) oven.
Fill each pita with lettuce, chicken, tomato, onion, and tzatziki. Serve while warm.