Hubris among the newbies - buying the legend that it is secure, when it is merely obscure. The basic security model is pretty much multi-user Unix, and not much better than Windows XP (assuming you actually use unprivileged accounts day-to-day)
Having a single administrator account (root) rather than a fully granular access control system as on e.g. VMS. So on a big system the admins are all-powerful and can do anything, when you might want e.g. to create operator accounts able to do backups and reboot the system but not hack the passwords. (you can get some of that function with sudo, I guess)
In practice, right now, common ways of hacking Linux are guessing user passwords over SSH and then looking for a privilege escalation exploit, and attacking Web 2.0 websites via SQL injection or XSS, which is more an application attack rhan anything to do with Linux itself.
In the past, there were issues with server processes all running privileged so that a remote exploit gave you root, but that's been fixed with chroot and privilege dropping so that servers can run as unprivileged users.