Whisk together first 8 ingredients until blended. Reserve 1/2 cup sauce to serve with chicken.
Place chicken in a heavy-duty zip-top plastic bag, and add remaining sauce. Seal and turn to coat chicken; chill 4 hours. Remove chicken, and discard marinade.
Grill chicken according to directions below. Serve cooked chicken with reserved 1/2 cup sauce.
For Gas Grills:
Indirect Cooking: Place 2 cups hickory, mesquite, or other wood chips in the center of a large square of heavy-duty aluminum foil; fold into a rectangle, and seal. Punch holes in top of packet. Preheat one side of grill, leaving center empty, for 20 minutes. Place packet on cooking grate over unlit side. Grill, covered with grill lid, 2 hours and 15 minutes or until done (170°). Baste as directed.
Direct Cooking: Preheat grill over low heat, under 300°, for 20 minutes. Place chicken, skin side up, on cooking grate. Grill, covered with grill lid, over low heat 1 hour and 15 minutes or until done. (Don't turn chicken.) Baste as directed.
For Charcoal Grills:
Indirect Cooking: Soak 2 cups hickory wood chips in cold water for 30 minutes; drain. (Wood chips for the gas grill are not soaked with water because they're encased in foil and placed on the cooking grate.) Prepare fire by piling charcoal on each side of grill, leaving center empty. Let charcoal burn for 30 minutes or until flames disappear and coals turn white. Sprinkle chips over hot coals. Arrange chicken, skin side up, on cooking grate in center of grill (not directly over coals). Cook covered with grill lid, for 50 minutes to 1 hour or until done. (Don't turn chicken.) Baste as directed.
Note: For these methods, we prefer to use a 3-pound chicken whole chicken cut into quarters, which tend to cook more evenly. If using chicken pieces, remove drumsticks and wings from the grill a little earlier to keep them from burning.