Put all the marinade ingredients into a bowl and pour over a generous amount of olive oil. Squeeze everything together with your hands to blend all the flavors.
Have the butcher cut the turkey into 4 pieces and remove the bones from the breast. (Save the neck and backbone for the gravy.) To marinate, put the turkey on a large platter and pour on the marinade; turn the turkey over in the marinade to make sure it is well coated on both sides. Cover with plastic wrap and refrigerate for at least 2 hours or overnight.
Remove the turkey from the refrigerator about 1/2 hour before you are ready to grill it. Heat the grill to medium and wipe the grate with some oil. Wipe the marinade from the turkey and season it well with salt and pepper. Put the turkey on the grill, skin side down, and cook for 30 minutes. Turn the bird over and continue grilling, basting with more olive oil, until the juices run clear and the internal temperature of the thigh is 180 degrees F, about 1 hour total. Set aside, cover with foil, and let it rest for about 10 minutes before carving.