The built-in Mountain Lion notification system works with Gmail, even if you're not using the Mail app.
In your browser, go to your Gmail settings and enable desktop notifications.
I wasn't referring to a browser setting. I meant the settings for Gmail itself.
If you open your Gmail account in your browser, and then go to the Gmail settings under the gear icon, you should see a setting for "Desktop Notifications" under "General".
I love mailtab for Gmail. It has a little more functionality than what it seems you want, but it's totally changed the way I interact with my inbox.
Sorry, I should have been clearer. My default browser is Chrome but I've changed the settings for Gmail though when I sent an email to my Gmail account, I don't receive any notifications...
I'm at a loss as to why it wouldn't work for you. This may be an obvious question, but are you running 10.8?