Preheat your grill or grill pan over medium-high heat. In a bowl, combine olive oil, oregano, garlic powder, onion powder, paprika, salt, and pepper. Coat the chicken breasts with the marinade and let them sit for at least 15 minutes.
While the chicken is marinating, prepare the tzatziki. In a mixing bowl, combine Greek yogurt, crumbled feta, lemon juice, minced garlic, dill, salt, and pepper. Mix well and refrigerate until ready to use.
Grill the marinated chicken for about 6-7 minutes per side, or until fully cooked and the internal temperature reaches 165°F. Remove from the grill and let it rest for 5 minutes before slicing.
Warm the pita breads on the grill for about 1 minute on each side.
To assemble the gyros, place a generous amount of mixed greens on each pita, followed by sliced chicken, cherry tomatoes, cucumber, and red onion. Drizzle with the creamy feta tzatziki.
Roll the pita tightly and serve immediately.