When it is really cold out especially at high aerobic rates - breathing hard can be a source of a source of dehydration, hypothermia as well as the upper bronchial irritation mentioned. Not usually a problem for a brief trip, but if you do it regularly or for longer distances it realy helps to breathe through something. anything from a balaclava over your nose and mouth to an Iditarod style re-breather will help. I recently bought a rather silly-looking gadget called the "Breathe" smog mask. purportedly for filtering smog, it works great at keeping the face, and more importantly the breath, warm.
HOWEVER, you will have quite a bit of moisture condensing in the mask which is also mildly unpleasant. On balance, I enjoy my winter riding much, much more with a warm face and warm air to breath in.
Special bonus!! you can get some kick-ass icicles forming on the outside vents of one of these which make you look like sir edmund Hillary.
Dork - factor - High due to ******** dragon graphics on the front.