I discovered this completely by accident but I'm really thankful because S-Voice has been an absolute failure for me in so many ways. First, S-Voice doesn't work when I have my Plantronics Voyager HD Pro headset on. When you double tap, s-voice comes up and then quits out immediately so it is non functional. When it is connect to my stereo BT in the car, S-Voice cannot hear me and is completely non functional. Google voice search still works but you have to do several steps just to get to it which is a paint while driving.

So if you've had it with s-voice and wish you had the global voice search you get when you do a long press of the search button which doesn't exist on the S3:

1) Install "Bluetooth Launch" from the play store.
2) Open Bluetooth Launch and select "Voice Search".
3) Expand it out and select "Recognition Activity"
4) Exit.
5) Double click the home button.
6) Select "Always complete using this activity"
7) Select Bluetooth Launch.

You're done.

Now whenever you double tap the home button, the stock ICS voice search will run. People have been asking all over the internet in places like xda for a solution to this but nobody has found one. 

The only reason I found this is because I knew I could use this app to launch any app from my BT headset. Ironically, on the S3, it's totally busted and doesn't work from my headset. But then I double tapped the home button and it prompted me to select either S-Voice or Bluetooth Launch!!!! I couldn't believe it. 

Skip step 6 above it you always want to choose between S-Voice and Voice Search. Additionally, you can choose any app or activity. It doesn't have to be Google Voice Search.

BTW, this works just like it did when we had the dedicated search button, so you be inside any app and get the global voice actions/search from the home button *without* exiting your current app.
Credit goes to Sorka