Not sure how Mozy backs up your iTunes library, but you shouldn't have this problem if your iTunes library is backed up and restored per the instructions included with this link.
Have you re-authorized your computer with your iTunes account with iTunes?
I have re-authorised my computer, but not synced yet.
All Mozy did was to copy everything under C:\Users\Mark Moss\Music\iTunes\iTunes Music, to a backup folder of the same name on mozy.com. So when I restored my hard drive's data, the iTunes was put back.
Under C:\Users\Mark Moss\Music\iTunes\Mobile Applications on my (repaired) laptop, I now don't have any mobile apps listed (I guess I'd forgotten to back those up but in any case they're on my iPhone) so I'm concerned that if I replace my iPhone, I'll lose the apps, which I want to avoid.