Yes, most classic games originated outdoors: football, baseball, watermelon bowling. Heck, public parks are even popular places to play board games like chess and checkers. Why not build an outdoor game room instead of an open air kitchen or family area? It won't matter if things get rowdy or loud. If someone spills a smoothie or drops a slice of pizza on the floor, the world won't come to an end, either.
Innovative outdoor rooms are big business. What do you really need for a game room, anyway -- electrical service, protection from the elements and a few amenities like comfy chairs. If you've ever visited a ballpark or stadium to enjoy the great American pastime or to watch some serious athletes throw the pigskin around, moving your gaming outdoors will seem like a natural.