Those Xbox One owners currently testing the Preview version of the upcoming March Update now have an added feature - the ability grab and save screenshots.
Double-tapping the Xbox button on the controller then hitting Y (or saying "Xbox take a screenshot" if your Kinect is plugged in) will save a screenshot of whatever you're looking at. You'll also be able to set a screenshot as your background via the Upload App.
The update will roll out publicly next month - but check here for the updated feature listing.