Grilling chicken can be complex if you are trying to grill several different pieces, or the whole bird on your grill. Like turkey, chicken pieces come in light and dark and each type requires a different technique. To keep the breast tender and moist it requires cooking with a lower temperature. However, at the same time, you will want them to have that grilled sear that gives it that distinct look and grilled flavor.
To do this, you will want to sear the breasts over a hot fire. Then move them to a low fire to complete cooking. This is easy if you have an upper and lower grill rack on your grill. You would sear the breasts on the lower rack then move them to the upper rack, while you are searing other pieces. Another way would be to set one side of the grill to a higher flame, and the other to the lowest setting. The same thing can be achieved with charcoal, by putting a double layer of briquettes on one side of the grill and a single layer on the other.
To prevent the chicken from getting an overpowering smoky taste with a charcoal grill, try cooking the chicken with the lid open. This will require longer cooking times, but will help eliminate the smoky flavor. This is really not a real concern with gas grills because gas burns cleaner so you can grill with the lid down without ill effects. Same goes for electric grills.
Searing is complete when all the pieces show good grill marks and the meat has just started to turn white. If you are grilling chicken with the skins on, then the skin should just be starting to crisp and the fat will just be beginning to drain. Watch the chicken closely to avoid flare-ups and burning. When this has occurred you will then need to move the pieces to the cooler parts on your grill.
Chicken breasts should be cooked around 10 minutes then turned and grilled about 5-7 minutes more. Grilling with the lid down, the pieces should take just a minute or so less. Whole dark pieces such as the leg and thighs will take an additional 16 to 20 minutes to cook, while thighs or legs separated from each other should take just about 12 to 16 minutes. Of course times will vary but chicken is cooked when it reaches an internal temperature of 160 degrees F. (71 degrees Celsius), which also happens to be the best temperature for flavor as well. Before long you will be known by your great chicken you cook on your grill.