JP & muno, thanks. The data discs are not blank, as I said they can be read on another machine.
The DVD drive on my machine had never been used to read a DVD-R disc before, that is why I am not sure whether Nero can be trusted or not when it says my drive supports DVD-R/RW discs.
I don't have a DVD-RW in hand, so I can't tell whether formatting the disc could help or not (DVD-R can't be formatted, as suggested by muno).
I've heard that changing the book type of the discs to DVD-ROM might help. But then I was told MY Sony DVD RW drive does not support the trick

.
BTW, the drive was bought last year, probably manufactured the year before that. Do you think it should be considerred a early model? I just have no idea.
Thanks again for your time.