I found a solution to the problem using Chrome.
I opened Chrome and went into plugins (by typing "chrome://plugins/" into the address bar) and disabled both native client and adobe flash player. I then switched tabs and went to amazon.com and tried to play a video and up popped a message that I needed to install flash player. I went back to the Chrome plugins and enabled both native client and adobe flash player. Switching back to the amazon.com tab, the video played! It worked for other videos too.
This is a strange solution I admit and I doubt that it will work for other people having similar problems but maybe this will shed light on the underlying problem and some Adobe person can use it to help others.
Amazon Instant Video still does not work in my Firefox or Internet Explorer browsers.