I've had the same problem, and it's a very serious one, isn't it? Who the hell would want to use DirectCD to back up files when it can't even preserve the date-time stamps on them? You'd think the bug [ahem... "issue"] should have come up during beta testing and been corrected?
At least once I was told by a Roxio tech last year that this is "just the way it is" and that there was no solution. However, I've just downloaded and installed an update for Easy CD Creator 5 Basic, which upgraded the DirectCD component from 188.8.131.52 to 184.108.40.206; just a few minutes ago I copied a folder and files to a DVD, and to my surprise the original date-time stamps appear to have been preserved. I used drag-and-drop to do it, but assuming it's really been fixed I'd expect that all methods would benefit.
So, unless I'm hallucinating because of sleep deprivation, your solution may lie in installing that same ECDC 220.127.116.11 upgrade.