1 Oil and salt chicken pieces: Coat the chicken pieces with olive oil and sprinkle salt over them on all sides.
2 Prepare grill: Prepare one side of your grill for high, direct heat. If you are using charcoal or wood, make sure there is a cool side to the grill where there are few to no coals.
3 Sear chicken on hot side of grill, move to cool side: Lay the chicken pieces skin side down on the hottest side of the grill in order to sear the skin side well. Grill uncovered for 5-10 minutes, depending on how hot the grill is (you do not want the chicken to burn).
Once you have a good sear on one side, turn the chicken pieces over and move them to the cooler side of the grill.
If you are using a gas grill, maintain the flame on only one side of the grill, and move the chicken pieces to the cooler side, not directly over the flame. Reduce the temp to low or medium low (between 250°F and 275°F, no more than 300°F).
Cover the grill and cook undisturbed for 20-30 minutes.
4 Turn over and baste with barbecue sauce: Turn the chicken pieces over and baste them with with your favorite barbecue sauce. Cover the grill again and allow to cook for another 15-20 minutes.
Repeat, turning the chicken pieces over, basting them with sauce, covering, and cooking for another 10-30 minutes.
NOTE that timing will depend on your grill set-up, the size of your chicken pieces, and how cold your chicken pieces are to start with! If you are grilling smaller pieces of chicken on a charcoal grill, they may be done much earlier. The goal is to maintain a low enough grill temp so that the chicken cooks "low and slow".
5 Finish with a sear or remove from heat when done: The chicken is done when the internal temperature of the chicken pieces are 160°F for breasts and 170°F for thighs, when tested with a meat thermometer.
Or if you insert the tip of a knife into the middle of the thickest piece and the juices should run clear, the chicken is done.
If the chicken isn't done, turn the pieces over and continue to cook at a low temperature.
If you want, you can finish with a sear on the hot side of the grill. To do this, put the pieces, skin side down, on the hot side of the grill. Allow them to sear and blacken slightly for a minute or two.
6 Paint with more barbecue sauce to serve.