Marinate the chicken: In a large bowl, whisk together olive oil, garlic, lemon juice, oregano, paprika, cumin, salt, and pepper. Add the chicken strips and toss until fully coated. Let it sit for at least 30 minutes (or up to 4 hours in the fridge).
Make the Tzatziki sauce: Mix Greek yogurt, grated cucumber, dill, garlic, lemon juice, and salt in a bowl. Cover and refrigerate until ready to use.
Cook the chicken: Heat a skillet over medium-high heat. Add the chicken in a single layer (work in batches if needed). Cook for 4-5 minutes per side until golden brown and cooked through. Let it rest for 2 minutes before slicing into bite-sized pieces.
Warm the pitas: Lightly toast them in a dry skillet for 20-30 seconds per side.
Assemble the gyros: Spread Tzatziki on each pita, add lettuce, tomato, onion, chicken, and a sprinkle of feta. Fold or roll the pita to enclose the fillings.