Android: On the android phone we used clicking and holding on the filename (example.mp3) brought up a menu that allowed you to download the file. The file was then found by browsing to Apps->my Files -> Download, where you could rename it to something more memorable and play it. It was also found automatically by google music.
iPhone/iPad/iPod Touch: There is an app called Capriccio It's free - but there is a paid version for 69p which has no ADs. This allows you to import mp3s of a website onto your iphone/ipod...i just tested it on yoganidranetwork.org and it works. To download a file you need to go to settings->network setting->URL Download and switch this to on. Then go to Media Explorer (magnifying glass) ->capriccio File System ->Download (disk icon) and paste in the url of where the mp3 file is on the web (copied from the site see above). You need to use the default UI (as opposed to the new one which you can enable in settings) - so don't do that for now (though I'm sure this feature wil be accessible in there in the future.