Grant, I AM Czech (on my mother's side, Bohemian father) and I can tell you the only reasons for you to learn that language is to train police/guard dogs or because you want to communicate with spies. Czech was the lengua franca of the spy trade in the 60's precisely because so few people knew it or wanted to know it. The only language which sounds worse to speak is Dutch (and I am Dutch on my father's side). A lot of my relatives still live over there, but my uncle is the only one who has visited them so far. My immigrant grandfather had a horrible time learning to write English because there are no vowels when you're writing Czech. I applaud your desire to teach English there, there are several work-abroad websites promising good positions in the CR, but maybe you could go to the American University there instead, once you are already working?
There are two Czech kids in our Thalia fan club, a boy and a girl, but I would have to ask them which language they speak because some of the people speak German. Splitting the country into Slovakia and the Czech Republic also complicated things. If you are serious, PM me and I will contact my kids there