High Definition Multimedia Interface
In general, there are graphics cards with DVI-D and/or HDMI connectors. Note that if your graphics card only has VGA (i.e. analog) output, while there is no way to directly connect that card via HDMI to a display, your best option may be the Aluratek AVH100F VGA-to-HDMI adapter.
ATI graphics cards
Good support, mostly used DVI-HDMI cable to connect HDTV. Don't forget setup Underscan to remove black borders. Can be set that in /home/mythtv/.xinitrc:
/opt/bin/aticonfig --set-dispattrib=tmds2i,positionX:0 /opt/bin/aticonfig --set-dispattrib=tmds2i,positionY:0 /opt/bin/aticonfig --set-dispattrib=tmds2i,sizeX:1920 /opt/bin/aticonfig --set-dispattrib=tmds2i,sizeY:1080
nVidia graphics cards
There are many graphics cards that have one or two DVI-D connectors, e.g. the Asus EN9800GT TOP/HTDP. With a DVI-D-to-HDMI-adapter(-cable), you can connect those cards to the HDMI input of your LCD TV.
The Asus EN9800GT TOP/HTDP graphics card doesn't have any support for HDMI audio or for patching the SP/DIF output of e.g. the mainboard through to the HDMI cable.
The Intel X4500HD graphics engine supports HDMI 1080p output with sound without using the standard non-proprietary xorg driver. This makes an easy out-of-the box setup. You can't buy a graphics card with a X4500HD, but you can buy motherboards, desktops, and laptops using it. The MSI G41TM-E43 is an inexpensive motherboard that works out of the box: The boot sequence and grub display fine over HDMI, and X works without needing an xorg.conf. You may need to switch the audio output to use HDMI.