The game that got closest to the spirit of TOS was probably The 25th Anniversary game and it's sequel released by Interplay. Having the original cast provide the voices was the icing on the cake.

Possibly the biggest bummer in Trek history is that The Secret of Vulcan Fury game was never released. Fans have hoped that somehow the original cast voice tracks have been saved, (including DeForest Kelley's last performance as Dr. McCoy) and could be the basis for an animated "lost episode".

http://www.youtube.com/watch?v=mBx-kqrm_zg

http://www.youtube.com/watch?v=KZT_qg0jH0U

JEDIT:Tobar beat me to it! ;)