Marinate the chicken: In a bowl, whisk olive oil, lemon juice, lemon zest, garlic, oregano, cumin, salt, and pepper. Add chicken and coat well. Cover and refrigerate for at least 30 minutes, up to 6 hours.
Prep the veggies: While the chicken marinates, chop tomatoes, cucumber, onions, and greens.
Pat the cucumbers dry to reduce excess moisture. Set aside.
Cook the chicken: Grill: Preheat to medium-high. Grill 5–6 minutes per side for breasts (thighs may need a minute more), until the internal temperature hits 165°F/74°C.
Stovetop: Heat a skillet over medium-high with a drizzle of oil.
Sear 5–6 minutes per side until cooked through.
Rest and slice: Let chicken rest 5 minutes, then slice into thin strips or bite-size pieces. This keeps it juicy and easy to layer.
Warm the pitas: Wrap pitas in foil and warm in a 300°F/150°C oven for 5–7 minutes, or heat in a dry skillet 30–45 seconds per side until pliable.
Assemble the wraps: Spread 2–3 tablespoons tzatziki down the center of each pita. Layer greens, chicken, tomatoes, cucumber, onion, and olives.
Sprinkle with feta and fresh herbs.
Wrap it up: Fold the bottom up, then roll from one side to the other. For easier eating, wrap the bottom half with parchment or foil.
Serve: Enjoy immediately. Add an extra squeeze of lemon or a drizzle of olive oil if you like.