I can, shame it doesn't support MKV properly, though!

Third party apps from the App Store, such as VLC, will play MKVs (and other formats) no problem.

I used to re-encode MKVs to MP4s for playback on the AppleTV 3rd gen, but since getting the Apple TV 4K (2nd gen), I simply shuttle over the MKVs to VLC from the web interface.

I'm interested - how do you transfer the 4K MKVs to the Apple TV?

In the VLC app on the Apple TV, you can open up the Network Sharing tab. It starts a local network server and is accessible via a webpage like http://living-room-apple-tv.local. Then I simply drag-and-drop the files from my computer to the webpage. On the AppleTV, they'll be cached indefinitely (at least until storage is available, or you delete VLC). I use this to play one-off/occasional video files.

You can also run a file server on your computer, and tell VLC where to access it over the network.

I believe that VLC isn't the only app on the AppleTV that can play MKVs, but it's the one I use.

In case you're wondering, the AppleTV is powerful enough to stream + transcode the MKV files on the fly. I've never had issues, at least with 1080p files. 4K should be fine, too.

Use Plex