You can experience Periscope broadcasts from your Apple TV (4th generation).
Start by downloading the Periscope app from the App Store on your Apple TV. There won't be an option to sign up for or login to an existing account, but you'll be able to view live and replay broadcasts as usual. You'll also be able send hearts by tapping the Touch surface on the Siri Remote and read comments from other people watching with you.
You can swipe down on the Touch surface to skip to the next broadcast, hide/show comments and hearts, and report sensitive content. Select the menu button to return to the main menu for an entirely fresh collection of places to go. If a broadcast you're watching ends, Periscope will automatically take you to the next destination.
If you notice that the borders of an image are drawn beyond the normally visible area, you can adjust the image by using overscan compensation on your TV or on the Apple TV Settings.