I usually smoke our turkeys lately. If your grill is big enough, it's easy.
1. Soak turkey in salt water for at least 8 hours prior to cooking.
2. Prepare grill: 20 minutes before cooking, stack 50 briquets and light. Take 2 lb bag of mesquite grilling chips and soak them in 1/2 bottle of red wine (inexpensive, merlot works well).
3. When a coating of grey ash has begun to form, separate into two piles of 25 briquets each, along both sides of a single use aluminum drip pan (under grill below bird). Add a handful of wine-soaked mesquite chips to each side. Set bird on grill. Place grill top over and set vents to allow a little smoke to release.
4. Every half hour (this is important - SET YOUR TIMER!) add 5 briquets and a small amount of chips to each side of the drip pan. Replace grill cover.
5. This will take approximately 4 hours for a 16 lb. bird, and 5 hours for a 20 lb. bird, etc. For a smaller bird, take at least 3 1/2 hours minimum to ensure complete cooking. Note that turkey meat may be pink, but juices should run clear.
Turkey can be brined with seasonings, if desired. A rub of rosemary and olive oil may be used, and cubed onions and apples added to body cavities will add flavor, but none of these steps are vital.
I have used this recipe with minor variations for at least 6 years, and it has never failed me.