If Xbox 360 Has Red Ring of Death Can I Still Transfer Files to Another Console?

I'm on my 3rd 360 console because the others died soon or later from the Red Ring of Death system failure error. I just started downloading all sorts of stuff after getting Xbox Live recently, so I'm wondering if the system I have right now crashes and breaks, is there a way I can transfer all the games I downloaded and payed for onto a new console's hard drive?

Suggestion:

You should be able since I heard that the red ring has something to do with overheating the processor which shouldnt effect the x box 360 hard drive.

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