My memory of the first time I heard that song is weirdly precise for something that happened (ugh) 22 years ago. The radio station I listened to most frequently back in those days was KRNA-FM, the rock station in Cedar Rapids. They'd been talking up the new R.E.M. album pretty heavily, and were particularly promoting their upcoming "world premiere" of the first single. It was a new sound for them, according to the DJs, very different from any of their previous releases.
So there I was in my room, listening to the radio, and finally the big moment arrives, and I listen, and I think, This could not possibly sound more like an R.E.M. song. You can make a case for Out of Time as a whole being something of a departure for the band, but "Losing My Religion" itself? Not so much. I think that's probably why the moment was so memorable; the reality of it was so far out of tune with how it was promoted.
Speaking of things that are out of tune, I happened to run across this odd little video earlier. It's "Losing My Religion" digitally transposed from a minor key to a major key. Strange!