The motherboard that I picked up for the VCR project provides out-of-the-box full-resolution HDMI video under Linux, but requires an additional proprietary Intel graphics driver to process audio through the HDMI port. Thankfully, this is not a terribly difficult process thanks to the fine folks at Intel providing an easy graphical installer package.
Head over to https://01.org/linuxgraphics/ and download the .deb package for Ubuntu.
Use your preferred package manager to install the .deb package, then run the installed package.
Follow the on-screen instructions to install the drivers.