I've recently switched to Mountain Lion, which means that I now have Notification Center.
I'm using the Twitter app by Twitter, and on my previous system I used Growl to display tweets. Now, however, nothing is showing up. Notification center does not seem to care much for regular tweets (without mentions), and suddenly Growl doesn't show them either.
A difference with my previous setup is that I downloaded the newest version of Growl now - it has evolved out of a prefpane and is now an app. I did confirm that it worked, though, as a Colloquy (IRC) mention did show up.
In the preferences of Twitter I did set 'Tweets' to show via Growl.