Cut tops off peppers and remove seeds. Place juice from tomatoes in a small dish. Place peppers in a small roasting pan. Combine onions, rice, mushroom pieces, ham and tomatoes. Spoon into peppers. Pour over the juice from the tomatoes. Cover and bake 350 degrees for 40-45 minutes. Sprinkle cheese over top and bake for an additional 5 minutes or until peppers are tender and the cheese is melted.