Marinate the Chicken: In a large bowl, combine olive oil, minced garlic, lemon juice, red wine vinegar, oregano, cumin, smoked paprika, ground coriander, black pepper, and salt. Add the chicken strips and mix well to coat evenly. Cover and refrigerate for at least 1 hour, but ideally 3-4 hours for maximum flavor.
Cook the Chicken: Heat a large skillet or grill pan over medium-high heat. Add the chicken strips in a single layer (you may need to do this in batches). Cook for about 4-6 minutes per side, or until fully cooked and browned. Transfer to a plate and cover to keep warm.
Warm the Pita Bread: While the chicken is cooking, warm the pita bread on a separate skillet or directly on the grill for a minute or two on each side.
Assemble the Gyros: Place a piece of pita bread on a plate. Add a portion of the cooked chicken. Top with tzatziki sauce, sliced cucumber, tomato, and red onion. If desired, add crumbled feta cheese.
Serve and Enjoy: Serve immediately while warm. You can also wrap the gyros in foil for easier handling.