1In shallow glass or plastic dish or heavy-duty resealable food-storage plastic bag, mix all ingredients except chicken. Add chicken thighs and drumsticks; turn to coat with marinade. Cover dish or seal bag; refrigerate, turning chicken occasionally, at least 30 minutes but no longer than 24 hours.
2Heat gas or charcoal grill. Remove chicken from marinade; reserve marinade. Place chicken, skin sides down, on grill over medium heat. Cover grill; cook 15 minutes. Turn chicken; brush with marinade. Cover grill; cook 15 minutes. Brush with marinade; cook 5 to 15 minutes longer or until juice of chicken is clear when thickest part is cut to bone (180°F). Discard any remaining marinade.