I've been streaming Amazon Instant Video for nearly 2 years on my Asus Transformer TF201, then suddenly, in the last few days, I was completely unable to stream anything from Amazon. Calling Amazon was unhelpful...the CS rep was rude and "smirky", saying basically "sorry about your luck, you should have gotten an ipad or Kindle.

I spent several hours searching for answers and finally got it working very well. I did try Dolphin, but the video was slow and pixelated, but I would have put up with it if there was no other workaround. I finally downloaded Boat. A lot of the instructions I read simply said to download and install the browser and that would fix it, but for me, I had to change a couple settings in the browser:
Install Boat
Go to Settings
Under "Page Content Settings" go to "Set user agent" and click "Desktop"
Then, under "Enable flash/plug-ins", set to "On demand".

This worked perfectly for me. I was able to stream, use full-screen with no problem and the video quality was quite good. (Actually better than it ever was with the standard browser.)

Hope this helps someone!