Zeroing out all bytes gives some level of security, but is not enough
against a truly determined adversary who has lots of resources,
according to what I've read.
A disk drive, being a mechanical device, will write to a slightly
different physical spot each time it writes to a particular bit position
on the disk. Sophisticated sensing mechanisms may thus be able to
determine what had been stored in that bit before a 0 was written to it.
For that reason, the more sophisticated shredding utilities do more than
merely write 0s; they will do so multiple times.