Steps :

1) Install Latest version of Snow Leopard.

2) Download any free app from Mac App Store.For example : Twitter



3) Go to your Applications folder in Finder, locate Twitter (or whatever free app you just downloaded), right-click, and click "Show Package Contents".

4) Navigate to the "Contents" folder and copy the following:

_CodeSignature
_MASReceipt
CodeResources

5) You will still need to find a way to download or acquire the Mac App Store .app files for the apps you would like to install (As I cannot condone stealing software, I won't be linking to the Angry Birds backup file you misplaced in this post, but if you truly did loose an app, chances are it's not terribly hard to find.

6) Right-click on the paid app file and click "Show Package Contents". Open up the "Contents" folder and delete _CodeSignature, _MASReceipt, and CodeResources. Now paste in the files you copied from the Twitter.app (or whatever free app you downloaded).

7) You can now play that Angry Birds app (and any other .app file from the new Mac App Store) to your heart's content. Just repeat the steps above.