As far as I can tell, it isn't possible to read from the SiI3114 RAID without a SiI3114 chip. I bought the expansion card linked in the question, installed it, hooked up the three drives, and it booted immediately.
Since this array is in RAID 5 and the total volume of data I want to preserve is small enough to fit on a single disk from the array, I plan to remove one disk, reformat it, and copy the contents of the array from the remaining two disks. I'll then reformat the other two drives individually and use them to make backups going forward. It's not as good a solution as I had hoped for when I setup the RAID, but it should be much easier to maintain moving forward.
I've had reasonable success with the approach outlined here. I have dual boot WinXP/Ubuntu running now on one disk from the old RAID. One problem I ran into was that Ubuntu got confused because, even after repartitioning and installing Windows, the drive still had RAID meta data on it. I could boot with a Live CD, repartition with GParted, mount and read the existing partitions, but the Ubuntu installer failed to recognize that the drive existed. I finally found a reference to the meta data problem and fixed it using dmraid. No idea why the Ubuntu installer is pickier than GParted, Disk Utility, and Windows Disk Management about the meta data.