How are you setting the game(s) to run at that resolution? If the game natively supports a resolution of 1920x1080 then there should be no black bars; however, if the game doesn't natively support 1920x1080 then you're going to have to use other methods of fitting the game to the screen. While it's a common resolution for TV's, it's not really that common for games. Your issue might be a very simple one.
WidescreenGaming has a list of all games and instructions on how to get those games to run in wide screen.
Master Games List - A - WSGFWiki