Prep the chicken: Pat the chicken dry. If using breasts, slice horizontally to create thinner cutlets for even cooking.
Make the marinade: In a bowl, whisk olive oil, lemon juice, garlic, oregano, cumin, smoked paprika, salt, and pepper.
Add chicken and coat well. Marinate for at least 20 minutes or up to 8 hours in the fridge.
Mix the sauce: Stir together Greek yogurt, lemon juice, olive oil, grated garlic, grated cucumber, and herbs. Season to taste.
Chill until ready to use.
Prep the veggies: Halve the tomatoes, dice the cucumber, and slice the onion. Chop olives and greens. Keep everything ready in bowls for quick assembly.
Heat the grill or pan: Preheat a grill to medium-high or use a cast-iron skillet/grill pan on the stove.
Lightly oil the grates or pan.
Cook the chicken: Grill 4–6 minutes per side, depending on thickness, until the internal temperature reaches 165°F (74°C). Let rest for 5 minutes, then slice into strips.
Warm the flatbreads: Place flatbreads on the grill for 20–30 seconds per side or warm in a dry skillet until soft and pliable.
Assemble the wraps: Spread a spoonful of sauce on each flatbread. Add greens, chicken, tomatoes, cucumber, onion, olives, and feta.
Finish with more sauce and fresh herbs if you like.
Wrap and serve: Fold the sides in and roll tightly from the bottom. Serve immediately while warm and crisp.