While you can use the Apple TV remote as a gaming controller by flipping it sideways, it doesn't provide quite the same experience as a dedicated game controller. However, you can get one for your new Apple TV.
The SteelSeries Nimbus is a gaming controller that you can connect directly to your Apple TV over Bluetooth and use it to play games. It has a similar design as the Xbox One controller, but with the D-Pad and left thumbstick switched around.
It costs $49, which is right on par with the cost of a game controller for an Xbox or PlayStation. It would be nice if users could use a PS4 controller with the Apple TV, since it uses Bluetooth as well, but we're guessing we'll see unofficial support for that arrive soon enough, either through jailbreaking or by other means.