Is there a way to stream MP3s stored on Amazon S3 via a Flash widget embedded in a website, or some other method?
To update the answer to this question, if you want to actually STREAM to clients, you can use Amazon Cloudfront on top of your S3 bucket (as mentioned by Rudolf). Create a "streaming distribution" in Cloudfront that points to your S3 bucket.
This will stream via RTMP (good for web and Android devices). You can use JW Player or a similar player to play the streamed files.
Cloudfront Streaming uses Adobe Flash Media Server 3.5.
There is also the ability to play secure content using signed urls.