I just visit put.io from my iPhone or iPad and from there I stream the video and send it to the Apple TV via Airplay. There's also a couple of free 3rd party native iOS apps that let you connect to your put.io account, and stream the videos, but they're not really necessary. Never had much luck with AirPlay mirroring with my Macbook Air, it's always laggy in my experience.

So yeah, you do need either an iPad or iPhone to do it, however Put.io has a Roku app if you have one of these.