Alternately thread tuna, onion, and mushrooms on 8 (12-inch) skewers. Place in a 13- by 9-inch baking dish.
Stir together hoisin sauce and next 3 ingredients; brush over skewers. Cover and chill 30 minutes.
Grill, covered with lid, over medium-high heat (350° to 400°) about 5 to 6 minutes on each side or until done. Serve with hot cooked rice.