I used to have the same issue when I tried connecting my GTX 260 to my 55" sony 1080p via regular VGA Cable. When I switched to using a DVI cable with a HDMI dongle at the end it filled the whole TV screen. Even when it filled the screen, it looked weird because I needed to go into my TV's video setting to move the screen around a bit and fill in the outside edges.
You might want to check out your TV's video setting. There might be something in there that allows you to adjust the screen.
Does your 3870 have a HDMI connection? Or are you using a HDMI dongle?
DVI TO HDMI DONGLE.
I will try to go to my video settings i guess.