Marinate the chicken. In a bowl, whisk olive oil, lemon juice and zest, garlic, oregano, cumin (if using), salt, and pepper.
Add chicken and coat well. Let sit for at least 20 minutes at room temp, or up to 8 hours in the fridge.
Make the tzatziki. Stir Greek yogurt with grated cucumber, dill or mint, grated garlic, lemon juice, salt, and pepper. Adjust to taste.
Chill until ready to use.
Prep the veggies. Halve cherry tomatoes, slice red onion, chop greens, and slice olives. If the onion is strong, toss with a splash of red wine vinegar and a pinch of salt for 10 minutes.
Cook the chicken. Grill, roast, or pan-sear. For a skillet: heat a little oil over medium-high, cook chicken 5–7 minutes per side (thickness varies) until internal temp reaches 165°F/74°C.
Rest 5 minutes, then slice thinly.
Warm the wraps. Briefly heat tortillas or pitas in a dry skillet or microwave for 15–20 seconds to make them pliable and less likely to tear.
Assemble. Spread a generous layer of tzatziki down the center of each wrap. Add chopped greens, sliced chicken, tomatoes, onions, olives, and feta. Finish with a sprinkle of sumac or smoked paprika if you like.
Fold and serve. Fold the sides in and roll tightly from the bottom.
Slice in half for easy eating. Add extra tzatziki on the side for dipping.