Technology issue: data from a damaged hard disk

Hello guys,

My former hard disk had some mechanical problems. It wasn't "corrupted" at all, so to speak.
I backed up my data (audio/video/picture files) on a brand new hard disk.

Someone told me that when a hard disk is damaged, then files could be "corrupted", and thus you may be saving damaged files on the new disks. That's not the case, I GUESS, because my former hard disk had mechanical problems, and some of the files (which I checked) can be opened on the new disk without any problem.

How could I check all the files (they are A LOT), in order to see if they are damaged? I think they aren't.