They now have an html5 video player, which I think they just introduced a week ago. BBC is about the only site I've seen that requires flash video, but I'm sure they'll get with the program soon.
So far I know NPR and Spiegel (german tabloid). All these sites work fine without flash if you set your useragent to that of a iDevice. Somebody implemented the switch between html video and flash in a foolish way.
I don't even have it on my system. I wonder if it's some sort of UA detection thing? I tried appearing as Chrome 41 on Windows 7, but no luck. (firefox nightly on linux user here, with h.264/aac support provided by gstreamer)