Anyway, so the workaround is pretty simple. First, select the Playlist you want to copy and go to File -> Export Song List..., then save that file (XML format is probably best) wherever you want. Then open iTunes on the machine you want to copy to, and make sure you have "Copy files to iTunes Music Folder when adding to Library" checked in the Advanced section of your iTunes prefs. Then go to File -> Import..., and find the XML file on the remote machine (assuming you've connected the two) and import it. This will import that list of songs and automatically copy them from the remote computer.
I did hit one snag in this process, as one album refused to import...it gave me a "Cannot find files" error and I could not for the life of me figure out why (there were no special characters or anything), but other than that it worked very well.
[robg adds: In my testing, dragging and dropping from the latest version of iTunes worked just fine. I'm posting this as an alternative method of migrating a playlist of songs from one machine to another.]

