Step 1: Marinate the chicken: In a medium bowl, whisk together the yogurt, olive oil, lemon juice, minced garlic, oregano, cumin, salt, and pepper. Add chicken pieces and toss to coat. Cover and refrigerate for at least 30 minutes, or up to 4 hours.
Step 2: Cook the chicken: Heat a grill or large skillet over medium-high heat. Remove chicken from marinade, letting excess drip off. Cook for about 5-6 minutes per side, until golden-brown and cooked through (165°F / 74°C). Transfer to a cutting board and let rest for 5 minutes. Slice into thin strips.
Step 3: Make the tzatziki: While chicken rests, combine Greek yogurt, squeezed grated cucumber, minced garlic, olive oil, lemon juice, chopped dill or mint, and salt in a small bowl. Stir until smooth. Taste and adjust seasoning. Cover and refrigerate until ready to use.
Step 4: Assemble the gyros: Warm pita or flatbreads according to package directions. Spread a generous amount of tzatziki on each warm pita. Pile on chopped lettuce, diced cucumber, chopped tomato, and thinly sliced red onion. Add sliced chicken and crumbled feta cheese (if using).
Step 5: Wrap and serve: Gently fold or wrap the pita. Serve immediately with extra tzatziki on the side.