From sources here and here, it would seem that it is impossible, if not ill-advised, to try and transfer music to your 360 hard drive. You can play music from a few different sources though, which are talked about in both sources:
- Flash Drive
By putting the MP3 files on a flash drive, you can navigate to them with the 360 interface, and play them.
- Rip a CD
Apparently, and this is news to me, you can rip files off of a disc and onto the 360. What's interesting about this option is that perhaps, once the files are ripped off the disc, you could potentially look through the filesystem for where the files were placed. That would be a good bet on where to place MP3 files, though playing them back afterwards is an unknown.
- Stream through Windows
There is the option to stream music through Windows Media Player. I also seem to recall a Windows Media Center which would do largely the same thing, but it's been retired as far as I know.