Place vinegar, oil, lemon juice, and lemon pepper in a small bowl or measuring cup; mix well. Place chicken in a single layer in a large sealable plastic bag; add marinade. Seal bag and shake gently to coat. Refrigerate 15 minutes. Turn bag over, keeping chicken in a single layer. Refrigerate for an additional 15 minutes. (Note: If desired, can be marinated up to 24 hours.)
Heat gas grill, then brush grill with oil.
Place marinated chicken on prepared grill of heated barbecue. Discard excess marinade. Cover barbecue and cook for 8 to 10 minutes over medium heat. Turn chicken over. Cover and cook for an additional 8 to 10 minutes, until juices run clear when chicken is pierced with a fork. Serve hot.