You can play music tracks through Dropbox's web interface, but only one at a time, in a very minimalist player. DropTunes provides a great web front-end for your Dropbox-synced tunes, offering Flash or HTML5-based streaming and continual playback. It's pretty darned handy.
You'll need to log into Dropbox through DropTunes to fire it up, of course, but you don't need Dropbox installed on a machine to use DropTunes. Launch the site, then pick out your folder with your music stash in the left-hand folder list. If you're syncing your iTunes library with Dropbox, DropTunes could be your personal Cloud Drive, and it's an easy way to share tunes with fellow Dropbox users — share the folder, and they can instantly access it through DropTunes. When the developer gets around to adding keyboard shortcuts, it's a safe bet we'll have a hard time containing our hyperbole.
DropTunes is a free service that requires a Dropbox account to use. It supports Flash-free HTML5 playback, but doesn't specifically support iOS or Android devices — though you may have some luck in the browser.