I experienced the same problem with Mevio.com yesterday. I also subscribe to several old time radio podcasts so I unsubscribed from them. When that didn’t work, I just deleted all the episodes, whether I had already listened to them or not. I also wrote to iTunes Support.
When I signed on to iTunes this morning, I didn’t encounter any dialog boxes from Mevio.com asking for my name and password. Oddly enough, the podcasts I thought I had deleted are still there. I was also apparently able to successfully download a couple of new episodes from some of my other podcasts.
I also received a reply from iTunes Support. Basically, they recommended that I uninstall, then reinstall iTunes. (They also, for some reason, seemed to think I am using some form of Windows as my OS.) Here, for what it’s worth, is what they said:
“I am sorry to hear that you're experiencing issues with podcasts on your iTunes Store due to a third party website asking you to sign in continuously. Nothing to worry, Tom; I'm here to help you.
“I recommend that you attempt to resolve this issue by uninstalling iTunes and then reinstall the latest version. To do this, first ensure that you are logged in to an Administrator user account on your computer, then follow the instructions in the appropriate article:
“Removing and reinstalling iTunes, QuickTime, and other software components for Windows Vista or Windows 7
<a href="http://support.apple.com/kb/HT1923" target="_blank">http://support.apple.com/kb/<WBR>HT1923</a>
“Removing and reinstalling iTunes, QuickTime, and other software components for Windows XP
<a href="http://support.apple.com/kb/HT1925" target="_blank">http://support.apple.com/kb/<WBR>HT1925</a>
“After successfully uninstalling iTunes, please ensure you have upgraded to the latest Service Pack for Windows by installing all updates available to you on Microsoft Windows Update:
<a href="http://update.microsoft.com" target="_blank">http://update.microsoft.com</a>
“Once you have the latest Service Pack for Windows installed, download and install iTunes for Windows:
<a href="http://www.apple.com/itunes/download" target="_blank">http://www.apple.com/itunes/<WBR>download</a>
“After successfully installing iTunes, navigate back to Podcasts. Everything should be working well.”