I decided to remove what I thought was the problematic extra nVidia video card, since there are already two DVI outputs build-in.  The card turned out to not even be nVidia, so I don't know what was going on there.

I futzed with the BIOS to configure the primary video card, which is some new Intel card.  The lucid (10.04) support for it is lacking, but it was easy enough to pull in new drivers from the appropriate Ubuntu PPA repository:

controls@pianosa:~ 0$ sudo apt-add-repository ppa:f-hackenberger/x220-intel-mesa
controls@pianosa:~ 0$ sudo apt-add-repository ppa:glasen/intel-driver
controls@pianosa:~ 0$ sudo apt-get update
...
controls@pianosa:~ 0$ sudo apt-get upgrade
Reading package lists... Done
Building dependency tree       
Reading state information... Done
The following packages will be upgraded:
  libdrm-intel1 libdrm-nouveau1 libdrm-radeon1 libdrm2 libgl1-mesa-dri libgl1-mesa-glx libglu1-mesa xserver-xorg-video-intel
8 upgraded, 0 newly installed, 0 to remove and 0 not upgraded.
Need to get 3,212kB of archives.
After this operation, 25.2MB disk space will be freed.
Do you want to continue [Y/n]?
...
controls@pianosa:~ 0$

After a reboot, both monitors came up fine.

http://www.subcritical.org/running_ubuntu_lts_on_sandy_bridge/