Primecast is a £3 app for Android that puts Amazon Instant Video on your Chromecast by scraping Amazon's backend to show you a list of your purchased and rented videos. From there, you can add them to a playlist, which Primecast can then sling to your Chromecast. If you're a Prime subscriber, you can also dig into the Instant Video offerings, but they don't show by default so you have to go hunt them down in the "Browse" menu. A minor annoyance.
The other catch is that you have to plug your Amazon credentials into this thing which, as Android Police points out, should terrify you. Also it's only a matter of time before Amazon figures out how to shut this thing off, but it has to be better than just streaming a Chrome tab. [Android Police]