Combine all ingredients. Reserve 1/2 for serving.
Combine the chicken and the marinade in a ziplock bag. Leave it in the refrigerator for at least 20 minutes or up to 12 hours.
Prepare a fire with the heat concentrated on one side of the grill.
Place the marinated chicken on the grill away from the direct heat (save some of the marinade for basting and discard the rest). Cover and cook for about 15 minutes, then uncover and move the chicken directly over the heat. Grill uncovered, turning, until the skin is crisp and the meat juices run clear, 15 to 25 minutes.
Serve with sauce, and cucumbers if desired.