Two ways that my husband (the one who cooks a turkey on the grill every holiday):

Turkey on the Grill 1

Preheat the grill to low. Put the turkey in a shallow pan with a little water. Place a stick (1/2 cup) of salted butter (not margarine) in the cavity. Close the lid. In about 15 minutes, when the butter melts, baste the turkey with the butter. Baste the turkey every 15 minutes until it's done. Total cooking time should be approximately 15 minutes per pound.

Turkey on the Grill 2

Same as above, but also place 2 cans of beer, opened and upright, into the cavity. Yup, that's right--can and all! The beer will produce steam and keep the meat moist. I don't recommend drinking the can of beer after you take it out of the turkey.