What solutions are there for getting HD video?
The Spartan 6 FPGA certainly can drive HDMI/DVI signals directly, demonstrated in the Atlys board. I've personally driven 1920x1200 VGA from a Spartan 3A starter kit, but the signal wasn't very clean then. You're looking at very high bandwidth signals, however, so the boards must be designed carefully. 

If this is for a single hobbyist build, you might be better off using a premade graphics card with its own framebuffer, for instance a USB one. DisplayLink comes to mind.