Remove neck and giblets; reserve for other uses. Remove and discard excess fat. Rinse the bird inside and out and pat dry. Season body cavity with salt and pepper. Tie legs together and twist wing tips under back. Brush turkey with oil and lightly sprinkle with salt and pepper. Arrange turkey breast side up, in center of cooking grate. Place lid on grill.
Cook 11 to 13 minutes per pound or until an instant read thermometer inserted into the thickest part of the thigh not touching bone registers 180 degrees and the juices run clear. Transfer turkey to a platter. Let stand for 15 minutes before carving.
Method for the Charcoal Kettle Grill: Open all vents on the grill. Use charcoal dividers to hold the charcoal briquettes. Divide the briquettes evenly, place them in the dividers and place the dividers on either side of the charcoal grate, as close as possible to the outside edges. Ignite the briquettes and, keeping the lid off, let them burn until lightly covered with a gray ash, about 25 to 30 minutes. If necessary, use long-handled tongs to rearrange the briquettes so the coals will burn evenly on both sides. Place a foil drip pan on the charcoal grate between the baskets of coals. Put the cooking grate in place, positioning the hinged sides of the grate over the briquettes so that more charcoal can be added easily. Arrange the bird breast side up in the center of the cooking grate. Place lid on the grill, leaving all vents open, and grill as directed above. Add briquettes as indicated on the chart below at the end of each hour to maintain the heat.
Charcoal Briquette Guide for Kettle Grill: For a 26 3/4-inch grill, use 30 briquettes on each side for the first hour of grilling and add 9 briquettes to each side every hour to maintain heat.
For a 22 1/2-inch grill, use 25 briquettes on each side for the first hour of grilling and add 8 briquettes to each side every hour to maintain heat.
For an 18 1/2-inch grill, use 20 briquettes on each side for the first hour of grilling and add 7 briquettes to each side every hour to maintain heat.
Method for the Gas Barbecue: Ignite the grill and turn all burners to high. Close the lid and preheat 10 to 15 minutes to bring the grill to 500 to 550 degrees. Turn burners down to medium heat. Place turkey breast side up in a roast holder inside a foil drip pan. Position pan in the center of the cooking grate and place the lid on the grill. For three burner grills, set front and back burners to medium and turn off center burner. For two burner grills, turn the front and back burners to medium. Keep grill covered.