Combine olive oil, oregano, garlic powder, salt, and pepper in a mixing bowl. Add chicken thighs (pounded to even 1-inch thickness if needed) and coat thoroughly with marinade. Cover and refrigerate for 30 minutes minimum, or up to 24 hours for deeper flavor.

Prepare tzatziki sauce by mixing Greek yogurt, crumbled feta, lemon juice, minced garlic, dill (if using), salt, and pepper until smooth. Taste and adjust seasoning as needed. Refrigerate covered until ready to use (can be made up to 1 day ahead).

Heat skillet over medium-high heat until a drop of water sizzles. Add marinated chicken thighs and cook for 6-7 minutes per side without moving, until golden brown crust forms and internal temperature reaches 165°F (74°C). Juices should run clear with no pink remaining.

Remove chicken from heat and let rest on cutting board for 5 minutes, then slice into 1/4-inch thick strips.

Warm pita breads in a dry skillet over medium heat for about 30 seconds per side until pliable and slightly toasted.

To assemble each gyro: spread 3 tablespoons tzatziki down center of warm pita, add sliced chicken strips, and top with cherry tomatoes, cucumber, and red onion. Fold bottom up first, fold in both sides, then roll tightly. Serve immediately.