Mix yogurt, garlic, oregano, and black pepper together in a bowl.
Add chicken to yogurt marinade, coating all sides. Cover and refrigerate for 4 hours.
Preheat an outdoor grill for medium-high heat and lightly oil the grate.
Remove chicken from yogurt marinade; place on preheated grill. Cook for 6 minutes; flip and sprinkle with feta cheese. Cook until chicken is no longer pink in the center and juices run clear, 4 to 6 more minutes. An instant-read thermometer inserted into the center should read at least 165 degrees F (74 degrees C).