In a mixing bowl, combine olive oil, red wine vinegar, dried oregano, garlic powder, salt, and black pepper. Whisk together until well combined.
Add the chicken breasts to the marinade and ensure they are evenly coated. Cover and let marinate for at least 15 minutes (or up to 2 hours in the refrigerator for more flavor).
Preheat the grill or skillet over medium-high heat. Cook the marinated chicken for about 6-7 minutes on each side, or until fully cooked (internal temperature should reach 165°F or 74°C).
While the chicken is cooking, prepare the tzatziki sauce. In a bowl, mix the Greek yogurt, grated cucumber, minced garlic, and dill. Stir until well combined and season with salt to taste.
Once the chicken is done, remove it from the heat and let it rest for a couple of minutes before slicing it into strips.
Warm the pita breads on the grill or in an oven for a few seconds until they are soft and pliable.
To assemble the pitas, place some lettuce or mixed greens in the center of each pita, followed by sliced chicken, cherry tomatoes, red onion, and a generous spoonful of tzatziki sauce.
Crumble feta cheese over the top for added flavor, if desired. Wrap the pita around the filling and serve immediately.