In a bowl, combine olive oil, oregano, garlic powder, onion powder, paprika, salt, and pepper. Add the chicken breasts and coat them well with the marinade. Let it marinate for at least 30 minutes (or up to 4 hours in the refrigerator for more flavor).
While the chicken is marinating, prepare the tzatziki. In a mixing bowl, combine Greek yogurt, grated cucumber, minced garlic, dill, lemon juice, and salt. Mix well and refrigerate until ready to use.
Preheat a grill or skillet over medium-high heat. Cook the marinated chicken for about 6-7 minutes on each side, or until the internal temperature reaches 165°F. Remove from heat and let it rest for a few minutes before slicing.
Warm the pita breads on the grill or in a skillet for about 1 minute on each side.
To assemble the gyros, place sliced chicken on each pita, then top with cherry tomatoes, diced cucumber, red onion, and crumbled feta cheese. Drizzle with tzatziki sauce.
Serve immediately and enjoy your homemade chicken gyros!