Yes, I do carry a travel mug - specifically one with a removable filter for loose tea. (I actually also have a model where the internal loose-tea-holding-pod can be retracted from the water with a little lever! I think that one's made by Aladdin.)

The university campus where I work, and the city I live in, are both extremely pro-reusable cup, and most places give you a small discount when you use one too.

(ETA Ugh it looks like this thread got killed, annoying.)