Marinate the chicken. In a bowl, whisk olive oil, lemon juice and zest, garlic, oregano, cumin (if using), salt, and pepper.
Add chicken and toss to coat. Cover and refrigerate for at least 30 minutes, preferably 2–4 hours.
Cook the chicken. Heat a grill pan or skillet over medium-high. Cook chicken 5–6 minutes per side, until nicely browned and cooked through (165°F/74°C).
Rest 5 minutes, then slice thinly.
Make the sauce. Stir Greek yogurt, grated cucumber, lemon juice, dill, garlic, olive oil, salt, and pepper. Adjust lemon and salt to taste. The sauce should be bright and creamy.
Prep the veggies. Slice tomatoes, onion, and cucumbers.
Chop lettuce and herbs. Keep each in separate containers to stay crisp.
Warm the wraps. For best texture, warm tortillas or pitas in a dry skillet for 15–30 seconds per side, or microwave with a damp paper towel for 20–30 seconds.
Assemble. Spread 1–2 tablespoons of sauce on a warm wrap. Add sliced chicken, lettuce, tomatoes, cucumbers, red onion, olives, and feta.
Add pepperoncini or herbs if you like.
Wrap it tight. Fold in the sides, then roll from the bottom up. Slice in half if serving now, or keep whole for meal prep.
Meal prep method. Store chicken, sauce, veggies, and wraps separately. Assemble just before eating for the best texture.