[Update: This workaround no longer works, unfortunately.]
In the comments to my rant about the geographical restrctions on the Amazon.com MP3 download service, Mark wrote that he could use it from Japan, using his mother's mailing address. This lit a lightbulb in my head (dim as it was): it seems that Amazon uses the default I-Click address to determine whether you have the right to download the MP3 or not. My default address was set to my Swiss address, so first I reset it to my U.S. address and tried buying a tune. No go still - I still got the rejection notice. So, I un- I-Click'ed all my non-U.S. addresses, and tried again. This time I was asked which address I wanted (I had two U.S. addresses to choose from). And bingo, I could download with impunity.