In a bowl, whisk together the yogurt, olive oil, lemon juice, garlic, oregano, paprika, cumin, onion powder, salt, and pepper. Add the chicken and coat well. Cover and marinate for at least 30 minutes, or up to 4 hours in the fridge.
In another bowl, mix the Greek yogurt, grated cucumber, garlic, lemon juice, olive oil, dill, mint or parsley, salt, and black pepper. Chill until ready to use.
Heat a grill pan, skillet, or outdoor grill over medium-high heat. Cook the chicken for 5–6 minutes per side, or until nicely charred and fully cooked through. Let it rest for 5 minutes, then slice into strips.
Warm the flatbreads or pitas in a dry skillet or briefly in the oven until soft and flexible.
Fill each wrap with shredded lettuce, tomatoes, red onion, and sliced chicken. Top with generous spoonfuls of tzatziki and crumbled feta. Finish with chopped parsley.
Fold and serve warm right away.
