App-synching and save-data question

[Jazzpha]

So, I finally made the leap to an iPhone 5 today, which was a pretty amazing change coming from a 3GS.

I synced my apps from my old phone onto my computer, and then put them from my computer onto my new phone-- only to discover that the syncing process didn't port over game-specific save data. Which means I have to do everything over again, from scratch.

Is there any way to transfer the save data from my old phone to my computer without jail breaking it, in order to correct this gargantuan kick to the nuts?

Thanks very much for your help!