Before last update it worked. Now I get a "unsupported browser" message.
You can switch to Google Chrome (not Chromium, since it doesn't include the Widevine content decryption plugin) and use native HTML5 player to watch prime videos. The only downside is you can't manually set the video quality to HD, but it's auto-set by determining your bandwidth.
FYI: I didn't update anything and it stopped working for me with the same message on 2015/10/21 (Arch Linux, FF 41.0.2).
After contacting Amazon, I was informed that since October 20 they no longer support Flash. They recommended installing Chrome (not Chromium), which will run the HTML5 player. That "solution" is working for me. Their support told me that currently no solution is available if you want to use Firefox in Ubuntu.