Make the chicken marinade: Cut the chicken breasts into uniform pieces - for example, in half crosswise, then cut the thicker piece in half horizontally. You can also pound the chicken in its thickest part with a meat mallet. If using chicken tenderloins, you can just remove the thick tendon in the middle (or leave if you don't mind). Mix all the marinade ingredients (lemon juice and zest, oregano, and oil), toss with the chicken pieces, season to taste with salt and pepper. Set aside to marinate for a minimum of 20 minutes or longer (a couple of hours or overnight).
Make the tzatziki: Peel the cucumber and grate it on the big holes of a box grater. Season lightly with salt, stir and let stand for 15 minutes. After this time, wring out the water as much as you can (just with your hands). Add the yogurt, pressed garlic cloves, and lemon juice to the cucumber. Season with salt (lightly) and pepper to taste and mix all ingredients together.
Cook the chicken: Heat the oil in a big frying pan / grill pan. When it's hot, remove the chicken from the marinade and cook on both sides until golden and cooked through (over high heat, this will take about 2-3 minutes per side). Cooking time will depend on the heat and the size of the chicken. Transfer the chicken to a plate and let stand for about 5 minutes before cutting into smaller pieces.
Assemble the wraps: Place one tortilla on a plate, spread about 1 heaped tablespoon of tzatziki sauce over it, top with a couple of chicken pieces, bell pepper strips, thinly sliced red onion, and strips of iceberg lettuce. Roll up the wrap. Repeat with the remaining ingredients.
Serve immediately and enjoy!