If you have a browser that supports Flash you can watch your videos on Amazon buy logging in to your account.
Amazon also have an app., which will allow Android devices to download videos for offline viewing but I seem to recall that it is not compatible with MM at this time.
Do a search for "Amazon instant video" to see if it is available for download.
I think Android Central recently did a short write-up on the App.
With the Amazon Video ap if you look under the hamburger menu you will see an option for downloads and that will also show the number of downloads you have on your phone. Tap on that and you'll have the options for your TV shows and movies that are downloaded on your phone.