It is probably bad form to reply to my own reply, but I received a quick reply from the very helpful folks at the
email address. Apparently there was an issue with the DNS for the site that hosts the MP3 files, so this problem should resolve within the next few days automatically.
In the meanwhile, if you want to patch it on your own machine (if you are impatient, like me!), assuming you are on Windows XP, you can add the following lines to the file C:\Windows\system32\drivers\etc\hosts on your computer (using Notepad or some other text editing program).
216.246.5.5 media.libsyn.com
38.100.193.81 media.libsyn.com
205.234.175.175 cache.libsyn.com
Give it a shot! That fixed it for me, hopefully it will for you as well.