[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.
I think it's still quite silly to reject me based on a shipping address. As Mark mentions, the iTunes Music Store bases their sendability on your billing address (I use a U.S. based credit card). I wonder why Amazon doesn't do the same. I only use U.S. based payment systems on the Amazon.com store. As far as they are concerned, I could be having stuff sent internationally to friends (and I have done that, as several non-U.S. addresses in my address book accumulated over the years reminded me).
Still, both are slightly more tolerable from my perspective than rejecting purchases based on IP address.
I know that the geographical restrictions themselves are not the fault of the vendors, but of the **AA organizations, who would rather bury their heads in the sand than face the realities of a globally accessible internet.