I'm trying to figure this one out too. Have my $300 Gateway laptop plugged into a $300 Seiki 39" LEDTV via HDMI and everyting works great... added a Logitech wireless keyboard and mouse and it's Linux nirvana.
For now I just made Mint's entry, which is like 4th on my menu list, the default one.
EDIT: I get text mode output from HDMI the second GRUB is out of the loop though, at 1080p no less, so I'm sure it's possible. It's same vid card so they just need to enable the port I guess. Shouldn't need any kind of new driver, that is.
picture of my setup: