Well, firstly, the drummer has to enjoy country music otherwise it's usually not a good experience for all involved :-)
I've played with a number of good rock drummers that just don't understand (or get) country and so they approach the songs just like normal rock tunes. That doesn't usually work out too well for the overall feel of the songs (although this is changing with a lot of the new country having much more of a pop/rock feel).
A country drummer needs to be very competent on playing shuffles, train beats, 3/4 waltz feel, able to use brushes effectively, half time feel as well as a few other beats that seem to be fairly specific to country.
To me a country drummer has to have almost the same mindset as a typical bass player....which is to lay back a little and just be subservient to the song and only play as much as is required to make the song perfect. Whereas in rock a lot of drummers take the approach to play as much as they possibly can get away with within a particular song (not that there's anything wrong with that).
I have played with a number of drummers that can play both rock and country very well. It's mostly just a mindset.
And, by the way, if you think country drummers just play rock beats, but only play them slower....then you should check out this song...
http://www.youtube.com/watch?v=xIzFo_UrD64
This is some pretty fast drumming if you ask me (and yes that's me on bass....hey, I gotta plug myself once in while)
:-)