Preheat your grill or grill pan over medium-high heat.
In a small bowl, mix olive oil, dried oregano, garlic powder, salt, and pepper to create a marinade.
Coat the chicken breasts with the marinade and let them sit for 15 minutes at room temperature.
Grill the chicken for about 6-8 minutes on each side or until fully cooked, with an internal temperature of 165°F (75°C). Remove from grill and let rest for 5 minutes.
While the chicken is resting, prepare the tzatziki sauce by combining Greek yogurt, lemon juice, and a pinch of salt in a small bowl. Stir well and set aside.
Slice the grilled chicken into thin strips.
Warm the pitas in the grill for 1-2 minutes until heated through.
Assemble the stuffed pitas: open each pita pocket and fill them with sliced chicken, cherry tomatoes, cucumber, red onion, some tzatziki sauce, and sprinkle with feta cheese and chopped parsley.
Serve immediately, with extra tzatziki on the side if desired.