Turns out the Linksys claim of Linux support wasn't quite as transparent as it implies (or I inferred). The card uses the tulip driver, but beware of the one installed with your RedHat distro. Depending on the RH version and the Linksys card version, you may have to recompile tulip.o. In my case, I had a version 5 card and RH 6.2. The card came with a floppy that included a utility for compiling and installing the proper files. Once I ran that, everything came up perfectly.
Guess I should've checked that before posting.