In a bowl, combine olive oil, oregano, garlic powder, smoked paprika, salt, pepper, and lemon juice. Add chicken breasts and coat evenly. Let marinate for at least 10 minutes at room temperature or up to 2 hours in the fridge.
Grate cucumber using a box grater, then wrap in a clean kitchen towel and squeeze out as much liquid as possible. In a bowl, mix the squeezed cucumber with Greek yogurt, minced garlic, dill, lemon juice, and salt. Stir well and refrigerate until ready to use.
Heat a large skillet or grill pan over medium-high heat. Cook chicken for 6-7 minutes per side, until golden brown and internal temperature reaches 165°F (74°C). Rest for 5 minutes, then slice thinly against the grain.
Warm tortillas or pita in a dry skillet for 30 seconds each side, or wrap in damp paper towels and microwave for 20-30 seconds. This makes them pliable and prevents cracking.
Lay each warm wrap flat. Spread 2 tablespoons of tzatziki in the center, leaving a border. Top with lettuce, tomatoes, red onion, sliced chicken, and feta if using. Fold in the sides, then roll tightly from the bottom. Slice in half diagonally and serve immediately.