Place chicken in a resealable plastic bag set in a shallow dish. For marinade, in a small bowl combine vinegar, oil, ketchup, 1 teaspoon of the barbecue spice, the molasses, mustard, Worcestershire sauce, garlic, and salt. Pour marinade over chicken. Seal bag; turn to coat chicken. Marinate in the refrigerator for 2 to 6 hours, turning bag once or twice.
Drain chicken, discarding marinade. Sprinkle chicken with the remaining 1 tablespoon barbecue spice.
For a charcoal or gas grill, grease grill rack. Place chicken on the rack of a covered grill directly over medium heat. Grill for 12 to 15 minutes or until chicken is no longer pink (170 degrees F), turning once halfway through grilling.