OK, this was a little frustrating. I downloaded the .iso and used the startup disk creator to make a usb flashdrive from which to install Ubuntu 12.04 Precise. Trying it out from the USB would work, and the installation would appear to work, but upon restarting I would get Error 1962: no operating system found (this after some weird stuff about trying to connect to MAC address...) Googling around I found other people have come across similar problems: http://askubuntu.com/questions/141879/error-1962-no-operating-system-found-after-installing-12-04-lenovo-thinkcentre
(Just FYI if anyone looks at this, messing with the BIOS: devices,
startup options, etc. did absolutely nothing to fix the problem. It
apparently has to do with the partitioning.)
Following their lead, I downloaded an older version of Ubuntu (10.04.4) from here: http://old-releases.ubuntu.com/releases/10.04.0/ choosing ubuntu-10.04.4-desktop-amd64.iso because I want to use the 64-bit version. I made a startup disk using this file, installed it on my hard drive, and restarted.
I'm not sure, but at this point I think I needed to remove my nvidea video card and just use the vga connector, but otherwise it worked fine. Then I ran update manager, waited for all updates to complete, then clicked the option to upgrade to 12.04. This took a long time, but in the end it worked. Restarting worked, and then replacing my video card and installing its drivers (using 'additional drivers' gui) got everything up and running! I would recommend this route. It takes time, but not much actual work. Just have a book handy while waiting for things to download and install.