I have this problem too and I don't believe it is an isolated problem but is widespread for Windows 7. I say this because I originally noticed this on my gf's laptop running Windows 7. And it is happening on my brand new HP computer too. I just downloaded Word
2013 and it is happening with the doc files of that program. It seems obvious to me that it is a Windows or Word bug. My autoarrange is off and it still happens. I never ever even changed that autoarrange setting, it was off by default on my new computer.
Update: Today, 2014-02-13, I noticed that a docx Word file icon moved on the desktop. I was surprised because according to the most helpful comment here (at this time) said the problem was only .doc related. In this case I had clicked on the save button and
I noticed that a temporary file was created on the desktop right when this happened.
Update 2014-02-15: Whether "Allow background saves" is checked or not does not matter - both doc and docx files still move occasionally though I have not been able to figure out what to do to consistently cause this to happen (in order to isolate the problem).
I've noticed the temporary file created is a ~WRLxxxxx file which according to one web page is a clipboard temporary file although the problem seems to occur even when I'm not using the clipboard.
I didn't want to use shortcuts because I thought it would make it harder to back up the files, but found that if I had an extra folder on the desktop to hold the documents that it would not make backing up any harder. But I shouldn't have to have an extra folder
on the desktop!
System
HP Z230 SFF computer
Windows 7 64-bit
Word 2013 32-bit (as recommended by Microsoft even for the 64-bit operating system)