Place the 2 chicken breasts in a bowl or resealable bag and coat with the chicken marinade; refrigerate for at least 30 minutes and up to 2 hours.
Preheat a grill or grill pan to medium-high heat.
Grill the marinated chicken 5–7 minutes per side, or until an instant-read thermometer inserted into the thickest part reads 165°F (74°C) and the meat is no longer pink.
Transfer the chicken to a cutting board and slice into thin strips.
Warm the pita breads briefly on the grill or in a dry skillet until pliable.
Assemble each gyro by placing grilled chicken strips on a pita, then add sliced cucumber, diced cherry tomatoes, sliced red onion, and crumbled feta.
Drizzle tzatziki sauce over the fillings, fold or wrap the pita, and serve immediately.