You should look into adding RBL (Blacklist) lookups to limit mail from known spammers. Additionally, I would modify the postmaster account in /etc/aliases to just junk all mail to postmaster:
That might not work for you, but I do not know many people that can say they'd rather deal with the spam than throw away mail for postmaster.
Perhaps if you gave us some information on the following that you are using we can help you lock it down a bit to limit Spam:
Linux Distro / Version
SMTP Mail Server