Amazon is now working on iOS devices. "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."
Of course, this implementation is still not as good as it is on Android, where Cloud Player is part of a native app. But if Amazon just did a little web work and made the web-based player optimized for the iPhone and iPad, it would certainly be very useable on a regular basis. Uploading, however, still requires Flash. But I assume most people are doing that from their computers anyway.
Amazon will probably announce the new compatibility at some point soon.