Question: I have an Asus EEE Transformer, and am contemplating an Amazon Prime membership. One of the nice features is streaming video. You can play it on computer, TV/DVD/BluRay that support it, or Kindle. 

Is there anyway to play on plain Android, like my Transformer?

Answer: 
Yes, you can play Amazon Prime videos using flash in the browser. See [this](http://forum.xda-developers.com/archive/index.php/t-1413969.html) thread.

Per the thread, you may have to remove the stock Flash and install Adobe's official version from the market, as well as the Dolphin HD browser.

You can rent movies from Amazon, which presumably use the same framework as Prime videos. It would probably be advisable to rent a movie to verify it works before investing in a Prime membership.
As of very recently, this no longer is possible. Flash for Android is no longer supported, upgraded or offered by Adobe.

There are hacks available for installing older versions of Flash, but given the security holes this opens up, installing an expired version of Flash is probably a really bad idea.