This post is a bit off-topic, but I could not find this information anywhere else so I decided to make a quick post.
Install the Amazon Unbox Video Player on your Windows PC
Then, go to Amazon.com Video on Demand
Pick a movie or TV show
Open the Amazon Unbox application you downloaded earlier
Wait for it to load
Go to Settings then Check for Recent Purchases
Your video will show
Right-click on the video and select Download to this PC
After it starts downloading it will actually be ready to watch. However, I usually wait until it is complete.
You can try to follow these directions to share the Amazon videos from your PC: http://www.amazon.com/gp/video/ontv/xbox, but on my Windows 7 computer I was not able to find “Media Sharing”.
Go to your X-Box 360 and select My XBox then Windows Media Center.
Note: You need to have your Xbox connected to your network and you need to set-up the Windows Media Center on your XBox.
In the Windows Media Center on your XBox, go to Pictures+Video
You will see the Amazon Unboxed folder
You will see your downloaded videos and you click on them to play them
This seems to only work for downloaded content, not the live streamed content. Therefore it is not useful for Prime members that want to use XBOX or Windows Media Center to play back the 'free' streaming videos that are now included in the Prime membership.