In a bowl, combine olive oil, oregano, garlic powder, onion powder, paprika, salt, and pepper. Add the sliced chicken and toss to coat. Let marinate for at least 30 minutes, or up to 2 hours in the refrigerator for more flavor.
While the chicken is marinating, prepare the tzatziki sauce. In a mixing bowl, combine Greek yogurt, sour cream, lemon juice, dill, minced garlic, and a pinch of salt. Stir until well combined. Refrigerate until ready to use.
Heat a large skillet over medium-high heat. Add the marinated chicken strips and cook for about 5-7 minutes, or until the chicken is cooked through and golden brown, stirring occasionally.
Warm the pita breads in a dry skillet or microwave for about 20 seconds until soft and pliable.
To assemble the gyros, place a portion of the cooked chicken on each pita. Top with cherry tomatoes, cucumber, red onion, and a generous dollop of tzatziki sauce. Sprinkle with crumbled feta cheese.
Roll up the pita to enclose the filling and serve immediately.