Place chicken pieces in a bowl. Mix vinegar with olive oil and pour over chicken pieces. Turn chicken pieces so that all are coated with the simple marinade. Let marinate for 30 minutes to an hour.
Prepare grill for medium high heat. Remove chicken pieces from marinade. Sprinkle generously with salt and pepper. Place chicken pieces on the hot grill, skin side up. Cook for approximately 30-40 minutes, covered, turning every 7 or 8 minutes. For the last 5-10 minutes of cooking, baste all sides with the honey mint mixture. The chicken is done when the juices run clear (not pink) when a knife tip is inserted into both the chicken breast and thigh, about 165°F for the breast and 180°F for the thigh. Note that although the breast pieces are bigger, they may finish cooking earlier than the thighs.