Go to Manage Apps, scroll down to Google Play Store, click on it and Clear data, Clear cache. You'll have to sign in again to Google Play Store but your apps you now be download able.

 

Your apps will still be there, listed under My Apps.

 

It didn't ask me to re-sign. It just asked me to re-confirm the store's terms. It caused some existing apps to get updated, but old apps have yet to show up.

So what's going on?

 

Assuming you are rooted, Titanium backup will allow you to backup your apps, and create a flashable zip file that you can flash onto your device (requires the Pro version of Titanium backup though).
If you are using a custom launcher, just backup the settings of that launcher to set up your home screen icons the way you want them.

 

In the play store app on your phone, go to menu--> my apps, and swipe over to "All" - you should see all the apps you have ever installed/purchased on that screen (assuming you are logged in with the same google account)

Manage apps - go to your phone's settings, go to the "All" tab, and click on "Application manager". Scroll down the list and you should see "Google play" app there. Clear the cache there. (I don't think it will ask you to sign in again by clearing cache though. If you clear data, it will ask you to sign in again)