At this point, the support Intel is offering for graphics controller is limited. Intel recommends checking the Linux software community to get proper drives for your onboard graphics controller.
If you are interested in additional information see below:
Thanks for the links. Asus also said that they don't formally support Linux for that board.
I haven't tried getting support from the Linux community yet because the relative priority of DVI vs HDMI seems to affect not only Linux and Windows operating systems but also the GRUB boot-loader, which presumably has a very basic generic driver built in. Even GRUB can display via DVI when HDMI is unplugged.
It seems that some part of the graphics system is automatically giving priority to HDMI over DVI, rather than vice versa, when the driver can handle only one. I can't find much info about the graphics architecture so don't know which component is doing this (on-die iGPU, Z68 chipset, mobo) but was hoping there might be a firmware fix or tweak that would reverse the priorities.
I would welcome any links to sites that explain graphics architectures, especially Intel HD Graphics 3000 / Z68. I might then be able to work out where the problem lies and perhaps avoid it in future boards.
It looks like I may have to buy a separate video card just to get dual output, which is a pity as the on-board graphics do everything else that I want.