Remove the neck and giblets from the inside of the turkey. Work the rub into as many areas as possible under the skin, as well as in the body cavity.
Prepare a grill for indirect cooking by layering soaked wood chips and charcoal. When the coals are ashed over, spread them out in one part of the grill so the turkey can cook to the side and not directly over the coals. Cover the grill and bring it to between 235°F and 245°F.
Smoke your turkey at 240°F for about 25 to 30 minutes per pound to an internal temperature of 165°F. (Cooking times will vary depending on your grill.) Cover with foil and rest turkey 15 minutes before carving. Serve with Stubb's Bar-B-Q Sauce.