In a medium bowl, mix together salt, black pepper, garlic powder, onion powder, oregano, paprika, olive oil, and red wine vinegar to create a marinade. Cut the chicken breasts into strips and add them to the marinade, ensuring they are well coated. Cover and refrigerate for at least 30 minutes, or up to 2 hours for more flavor.
Preheat your grill or grill pan over medium-high heat. Remove the chicken from the marinade and discard any leftover marinade. Grill the chicken strips for about 5-7 minutes on each side, or until fully cooked and the internal temperature reaches 165°F. Remove from the grill and let rest for a few minutes.
While the chicken is resting, prepare the tzatziki sauce by combining Greek yogurt, lemon juice, diced cucumber, salt, and pepper in a small bowl. Mix well and set aside.
Warm the pita breads on the grill for about 1 minute on each side, or until they are soft and pliable.
To assemble the gyros, place a few strips of grilled chicken on each pita. Top with diced tomatoes, red onion, parsley, and a generous spoonful of tzatziki sauce. Roll up the pita to enclose the filling.
Serve immediately and enjoy your homemade Greek chicken gyros!