Put the chicken, skin side up, over the unheated part of the grill, cover, and cook, turning once or twice, until the chicken is no longer pink and the juices run clear (about 170°F for breasts and 180°F for thighs on an instant-read thermometer inserted into thickest part without touching bone), 30 to 40 minutes total. Breasts cook faster, so check them first. During the last 10 to 15 minutes of grilling, move the chicken over the heated part of the grill to brown all over. Brush with the barbecue sauce during the last 5 minutes.