If you visit Amazon’s Cloud Player through the Safari web browser on an iOS device, you’ll see that it does in fact now work. You’ll first hit a warning page telling you that your browser is not supported, but just ignore that. Click into the music in your drive and it will begin playing. It works flawlessly — even to the point where if you get a Push Notification or incoming call, the music will be paused.
Amazon will probably announce the new compatibility at some point soon.