Marinate the Chicken: In a large bowl, whisk together the olive oil, lemon juice, minced garlic, oregano, paprika, cumin, salt, and pepper. Add the chicken thighs to the marinade and toss to coat evenly. Cover the bowl and refrigerate for at least 30 minutes, or up to 4 hours.
Prepare the Tzatziki Sauce: While the chicken is marinating, make the tzatziki. Grate the cucumber and then squeeze out as much excess liquid as possible. In a medium bowl, combine the grated cucumber, Greek yogurt, minced garlic, chopped dill, lemon juice, salt, and pepper. Stir well to combine. Taste and adjust seasonings as needed. Cover and refrigerate until ready to serve.
Cook the Chicken: Heat a large skillet or grill pan over medium-high heat. Once hot, add the marinated chicken thighs. Cook for 6-8 minutes per side, or until the chicken is cooked through and slightly charred. The internal temperature should reach 165°F (74°C).
Remove the chicken from the skillet and let it rest for a few minutes before slicing it into thin strips.
Warm the Pita Breads: While the chicken is resting, warm the pita breads. You can do this in a dry skillet, in the microwave, or in the oven.
Assemble the Gyros: Lay a warm pita bread on a plate. Top with sliced chicken, shredded lettuce, sliced tomatoes, and sliced red onion.
Spoon a generous amount of tzatziki sauce over the top.
Fold the pita bread in half and serve immediately. Enjoy!