Heat gas or charcoal grill. Cut 4 (20x18-inch) sheets of heavy-duty foil. Fold each in half to form 10x18-inch rectangle.
In large bowl, mix potatoes, mushrooms, carrots and onions. Crumble beef over vegetables. Add barbecue sauce; toss until well coated. Spoon 1/4 of mixture onto each sheet of foil. On each sheet, bring up 2 sides of foil so edges meet. Seal edges, making tight 1/2-inch fold; fold again, allowing space for heat circulation and expansion. Fold other sides to seal.
Place packets on grill over medium heat. Cover grill; cook 20 minutes, turning packets over once, until vegetables are tender and beef is brown. Carefully open packets to allow steam to escape.