Start by navigating using the Folders presentation on the bottom left of the Cloud Drive screen. It's the most effective way to move through your drive on your iPad or other iOS device without running into snags from Flash-based elements.
Instead of using play/pause, once you've navigated to your music, tap the check box next to the item name, and then tap Download. Mobile Safari connects to the resource and then starts playing the MP3 file back natively using an embedded player.
It may not be as lovely to look at as the unified music list GUI, but in our tests, we were able to play back a variety of music formats, not just MP3.
I went ahead and purchased a copy of Natalie Maines' new "God Only Knows" track from iTunes, which downloaded as an m4a MPEG-4 Audio Layer file. When tapped, it completely downloaded, and then presented the following "Open in" options. I went ahead and opened it in VLC, where it immediately started playback.
Unfortunately, the complete download/transfer-to-app process sort of kills the whole "Streaming from the Cloud" paradigm. It can occupy a lot of space on your device, and you cannot add the music to your iTunes library. (Well, to be honest, yes, you can, but not through any Apple approved methods at this time.) You may want to delete the downloaded files after playback. Be aware that some apps handle "Open in" functionality and the system Inbox better than others.
If you stick to MP3 files, however, you don't have to worry about "Open in." The files play back in Safari and you can tap the "Back" button to return to your Cloud Drive homepage. You still have to wait for them to download to play back, but at least your music will be available for you to listen to wherever you are.
Pro tip: When purchasing music, add it to your Cloud Drive first and only then download it from your Cloud Drive to your computer. When you download immediately, you lose the "add to your Cloud Drive for free" option.