Media Center does not automatically move files. Assuming you're not leaving out a substantial number of intervening steps, something else is moving them, or your filesystem is horribly broken.
Check the disk for corruption:
http://wiki.jriver.com/index.php/Troubleshooting_DisksIf chkdsk finds all sorts of errors, then you may have larger issues. If that comes back clean, then search for them in Windows Explorer and find out where they're being moved. That might give you a clue as to what application is moving them.
iTunes does automatically reorganize files, so it could be iTunes.