In blender, place all marinade ingredients. Cover and blend until smooth. If turkey has plastic leg holder, remove and discard before grilling. Inject marinade into turkey, following the directions that came with the injector. (Or pour marinade over turkey on large tray with 1-inch sides or in large glass dish.)
Sprinkle Cajun seasoning inside cavity and over outside of turkey. Fasten neck skin to back of turkey with skewer. Fold wings across back of turkey so tips are touching. Tuck legs under band of skin at tail. Place turkey on large tray with 1-inch sides or in large glass dish. Cover and refrigerate 8 hours but no longer than 24 hours.
If using charcoal grill, place drip pan directly under grilling area, and arrange coals around edge of firebox. Heat coals or gas grill for indirect heat.
Insert digital oven-cord thermometer in turkey so tip is in thickest part of inside thigh muscle and does not touch bone.
Place turkey, breast side up, over drip pan or over unheated side of gas grill. Cover and grill over medium heat 3 to 4 hours or until thermometer reads 165°F and drumsticks move easily when lifted or twisted. Let stand 15 minutes before carving.