The way you describe above IS the "long and tedious" way of doing it ... but try this instead:
Just go into your settings, open the Application Manager, then start tapping on the apps you want to stop the notifications on ... when you tap on the app, it pulls up the screen you show above (for that particular app, of course) ... then just uncheck the "show notifications" box.
Yes, it is still a little tedious, but that's the most efficient way I know of to do it. Just be sure to not only check the "downloaded" apps, but page-over to the "running apps", and/or "all" apps to make sure you don't miss any of them that you want to turn off.