Ok, so I just found a picture of the app on the various devices (http://www.chess.com/mobile-chess.html) and the picture of the app on the Android phone looks *NOTHING* like it does on a Kindle Fire. So now I'm thinking maybe the app that is in the Amazon app store is not the same as the one in the Android market?
Android app vs iPhone/iPad app
...that's exactly what I was about to speculate.
Kindle Fire does indeed run android, but Amazon has covered it up with it's own operating system that runs apps differently than other devices. The Kindle app store is also limited, and doesn't offer the same apps that Google does for most android devices.

I wonder if I should try to snag a copy of the apk file from the Android market and see if it plays nice.

Maybe you should let erik give his lecture on this.
Is that a good thing or a bad thing? :-)

I wonder if I should try to snag a copy of the apk file from the Android market and see if it plays nice.
Try these links:
http://www.chess-3.com/androidlive.apk
http://www.chess-3.com/android.apk
I can also send you several versions, including the last one. Send me a message with any valid email.

Ok, so I just found a picture of the app on the various devices (http://www.chess.com/mobile-chess.html) and the picture of the app on the Android phone looks *NOTHING* like it does on a Kindle Fire. So now I'm thinking maybe the app that is in the Amazon app store is not the same as the one in the Android market?
Just a note, the picture up on that page for Android devices doesn't look like the app on my phone either. Maybe earlier versions looked like that but not the one I'm on. The game screen (for Online chess at least) doesn't have any of the navigation buttons on top or bottom. If you want the ones on the bottom I have to hit the Menu button to get them.
The Home screen and tactics look essentially the same as the screenshots here, with the exception of the bottom row of buttons on the tactics screen (again need to use the menu button of the phone to access for me). So there are differences between the screenshots and the current application, at least for me.
This is on an Android phone, though I would assume the UI is essentially the same across the hardware platforms.

Martin, what you're describing sounds pretty much exactly how it is on the Kindle Fire. I would have definitely expected some differences between the iPhone and Android versions, but it's a little more than I would have though. Oh well, I'm getting used to it. I still prefer the iPhone version far more than the Android one, but am indeed grateful to have it on both.
I've been using the app for the iPhone for about a year and a half and have absolutely loved it. I got a Kindle Fire today and installed the Android app. What a disappointment! This app feels archaic compared to the iPhone version. Moving to the analysis screen or any of the other screens from within a game is quite cumbersome in comparison. Why is there such a disconnect in how this app is developed compared to the other one? I realize they are probably developed by two different dev teams, but I'd think they'd be consistent.
Any future plans to bring them into alignment with each other (hopefully the Android app moving toward the same user experience as the iPhone app)?