They would still be able to post but their posts wouldn't bump the thread to the front page. And requiring to post in the ANN thread is already against the rules.
Although I still prefer stompix's idea of having to spend merits to bump a thread. This would cost abusers a lot more than just getting a merited account and using it for unlimited bumps.
Alternatively, we give the owner of the thread a bump button which they can do at any point allowing them to bump their threads strategically or when they are online. Don't allow posts to bump threads anymore. Only allow the button to appear every 24 hours.