I'm not convinced that running specifically should be a required part of martial art training but physical conditioning should be. Martial Art is a physical endeavor and being strong, agile, and having good cardio can only improve one's ability to perform their art. Personally, I weight train three days a week and do cardio on the other three days. In class we open with a bit of calisthenics.
I don't have specific fitness requirements for belt promotions, however the tests are difficult enough that most of my students do outside fitness training in order to perform.
The above is just my view on things, people train for a variety of reasons and should train acordingly.
Mark