Soak the wood chips in a bowl of hot water to cover; let stand for at least 1 hour and up to 5 hours. Drain, then transfer to a wood smoking box. (Alternatively, wrap a 5-by-7-inch disposable aluminum pan with foil and poke about 10 holes in the top.)
Remove the grate from a gas grill and place the wood-chip packet directly over the burners on one side of the grill. Position a shallow baking sheet for a drip pan on the other side of the grill; replace the grate.
Preheat the grill over medium heat until smoke begins to escape, about 10 minutes. Rub the turkey with olive oil and season with salt and pepper (omit the salt if the turkey is brined), including the cavity. Stuff the cavity with the aromatics, if using; tie the legs together with kitchen string. Brush the grate positioned over the drip pan with oil. Arrange the turkey, breast side up, on the grill rack and pour 2 cups water into the drip pan; cover the grill with the lid. Lower the heat to low so that the temperature reads about 350 degrees .
Cook the turkey, undisturbed, until an instant-read thermometer inserted in the thigh registers 155 degrees to 165 degrees , 2 to 2 1/2 hours, adding more water to the pan if needed. Tent loosely with foil for 15 minutes before carving.