I did a search when I got my new Evo 4G and all I found was people saying it wasn't possible and the ones that said it was possible said you had to be rooted. After a little more digging and some trial and error I figured out a way to backup all THREE versions of Angry Birds and then move all progress (including golden eggs) to a new and UNROOTED device for FREE! Here's how I did it. Hope this helps others.
There are free apps in the market that will allow you to backup and move your progress to another device w/o rooting. I wanted to reply because I went through the same thing on my new device and I had three devices actually. I had Angry Birds on my Droid 1 (rooted), Angry Birds Seasons on my tablet (not rooted) and finally Angry Birds Rio on my new Evo 4G which wasn't rooted. If you search the market you will find these three apps (Devs listed on right) to be the best as they backup not only progress but golden eggs and what not and are all free:
angrybirdsbackup - ANTBEE
angrybirdsbackup (seasons) - ANTBEE
Rio backup - Whitney Powell
The first two are from the same dev and the third seems to be about the same thing from a different dev. It would be nice if the same dev offered one for all three games or, even better, if that dev would COMBINE them all into a single app *hint hint* that generated separate folders for each game the end users chooses to back up.
Here we go:
1) Install whichever you need on both devices.
2) Complete ONE level on the new device (do NOT skip this step or you won't be able to complete the process successfully) and then create a backup to your device using the app.
3) Install the same app on the old device and create a backup for the same game on the old device. Now you have backups of current progress of the same game on both devices and you need to get the old to the new.
4) Use a file explorer (I like ES File Explorer best out of all of them but some prefer Astro... either will work for this DIY) to locate the folder that was created on the new device which will be named the same as the backup app. Within that folder you will find two files which I think are .bkp or something similar. One is scores and one is something else. You want to delete those two files from the new phone and then copy the same two files from the old phone and place them in the same location on the new phone.
5) Back out, launch the backup app that you installed and select restore.
You can verify by launching Angry birds to see your progress is all stored on the new device. Making a backup of your Angry Birds games every once in a while and then copying those backups to a PC or external storage device isn't a bad idea if you're serious about Angry Birds.
I see this question asked often but rarely see an answer nor have I seen a step by step on how to complete it even if you're able to stumble upon one of these apps since they don't include instructions.