Amazon Kindle forum support (source)

With the announcement of Google Music and of course having multiple Android Phones I wanted to keep my music in one location.  With the help of some Google searches I found some steps on the Amazon Kindle forums that worked perfectly, modified slightly for what I needed to do in order to get it working.

1. SideLoad: GoogleServicesFramework.apk and Books.apk
2. Reboot the Kindle Fire
3. Start the book Application and set it up to sync with your google account.
4. Once the sync is complete, Reboot the Kindle Fire again
5. Side Load Music.apk
6. Start music application and it should sync
7. (optional) uninstall books, I left it, never know if Google will get something good and it is small enough anyways

So far in my experience after the initial sync it does not appear to auto sync.  If you experience this also, you can go into settings and force a re-sync.