In a large bowl, whisk together olive oil, red wine vinegar, lemon juice, minced garlic, oregano, salt, black pepper, paprika, and cayenne pepper (if using). Add chicken thighs to the marinade, ensuring they are well coated. Cover and refrigerate for at least 1 hour, or up to overnight for best flavor.
Preheat your grill or grill pan over medium-high heat. Remove chicken from the marinade and discard the marinade. Grill the chicken for about 6-7 minutes on each side, or until fully cooked and the internal temperature reaches 165°F. Remove from the grill and let rest for 5 minutes.
Slice the grilled chicken into thin strips.
Warm the pita breads on the grill for about 1 minute on each side, or until soft and pliable.
To assemble the gyros, place a generous amount of sliced chicken on each pita. Top with diced tomatoes, shredded lettuce, sliced red onion, and a drizzle of tzatziki sauce.
Roll up the pita to enclose the filling and serve immediately.