Boy, that was tedious. But the problem is now gone.
Today is the first time in two weeks I have had enough free time to sit down and pick through all my podcasts to check the URLs for Mevio-relatedness.
I copied the podcast URL for each and every podcast into a TextEdit file. Not a single one of them had the word mevio in the URL.
However, I took this opportunity to delete a bunch of podcasts that I have kept in my iTunes list, but have either unsubscribed or else the podcaster has long since quit updating the feed. It was a tiny bit emotional, since I was dumping podcasts by the original (or second-wave) podcasters from 2005, like myself. But I was cold-hearted about it, and I deleted everything other than my current podcasts (and the one I produced all those years ago).
Among those were a couple of Adam-Curry-produced podcasts from many years ago that did NOT have mevio in the URL. Nevertheless, once I dumped those, the problem seems to have gone away.
So I hope my own Mevio-iTunes nightmare is now over.
The real message I want to convey here is this: copying out the URLs into a text file and looking at them might well not do you any good at all. It might not be at all obvious what the offending podcast is. You may well have to simply dump those casts and then go look for them again in the iTunes store or on the Interwebz somewhere.