I am still waiting when I can play Netflix on Linux.
see this: http://www.amazon.com/b/ref=sa_menu_aiv_piv_t10?ie=UTF8&...
Edit: According to rglullis, this is wrong. I have left it for context. Sorry - I had read something about it, but it appears to not be true.
I "installed" the app, which only took me to the website. I selected a movie and got a page showing their requirements page. It mentions "PC", Mac and Chrome OS.
No love for linux, at least not right out of the box.
In the mean time, buy a Roku and plug it into one of your monitor's other outputs.