Solution
The app you want is called Air Video for iOS. I swear by it and have absolutely no affiliation with the developer. It does on-the-fly transcoding from any Mac to iOS, and supports AirPlay. You can also queue transcodes for future playback. You just have to install their free server client on the Mac and you're good to go. I'm using a 2009 Mac Mini and it plays 1080p MKV movie files with amazing quality. You can even stream over the internet and it looks pretty good.
I just bought a new house and there's no place for the Mac Mini under our TV on the wall, so I'm looking for a solution like this myself. I found this thread through a Google search, and as I was reading it I remembered Air Video is already a working solution.
If you are taking the Mac completely out of the equation (aside from downloading the MKV files) then I think VLC might support AirPlay... but you might run into weird compatibility issues.
Air Video for iOS is what you want, trust me. Looks like there is a new version I didn't know about... I guess it wasn't a free update. hmmm...
http://www.inmethod.com/index.html;jsessionid=11AEA4DC00F40B79C6D8CD207C01FF9A