I researched this topic briefly
and found this: http://gear.ign.com/articles/693/693530p1.html
The blurb of note:
Datel cautions, however, that some saves, demos, and downloads are locked to your particular 360 or online account. Console locked files won't work with any other console, and some other files can only be used with a particular Xbox Live account. Datel says that most saves and demos are totally open, so don't worry too much
So....depending on the game, you should be able to transfer at least some of the locked files if they came from your original xbox with your live account that is the same on both machines (but that is assuming some things about the digital signature and how its generated).
It would be great to see an actual list of games that are locked and what method they use but I couldn't find one.
As for getting the saves from your own xbox (hopefully getting at least most of the "locked saves" if they are locked to your actual account and not the console itself), the process is described here: http://us.codejunkies.com/news_reviews.asp?c=US&cr=USD&cs=
It looks like you would need Action Replay for the original xbox as well as the one you have already.
If you do go this route, I would really like to know if it works for you and which games it works on. Stating the obvious but if only MS would automate this through live.