I had trouble too but was finally able to get it installed through a very round about way. Once I finally got the appstore running, it downloaded Angry Birds Rio but it wouldn't install (with the other versions of Angry Birds, the SD card needs to be unmounted to install). Since Amazon saves the apps to the SD card, I couldn't unmount the SD card to install it.
So I had to plug the phone into my computer, find the APK file (sdcard/Android/data/com.amazon.venezia/cache) and move it to the internal memory. Then I unplugged the phone from the computer, opened into Astro file manager on the phone, went into the phone's internal memory, found the file and installed it from there, with the SD card unmounted. I remounted the SD card and Angry Birds Rio works with no problems.
There may be a simpler way to do this but this way worked for me. I hope this helps.
Just figured that I would pitch in a work around I found as it may help some....
My wife was so jealous that I was playing Angry Birds Rio on my 'Bolt that she had forbidden me from using my phone until I fixed hers . We already had an Amazon 1-Click account setup with a credit card and it still didn't work. All I had to do was power her phone and pulled out the MicroSD, powered it up and downloaded and installed the Amazon Appstore App, shut down when finished, popped the MicroSD card back in and powered up the phone, finally downloaded Angry Birds Rio, enjoy!
I hope this helps, now finally back to my 'Bolt heh.