It might be hard to motivate yourself to exercise out in the elements on a freezing day, but your body can handle the cold.
Afraid your will lungs freeze from the cold air? Your body is designed so that by the time air reaches your lungs, it's been warmed up and moisturized by your nasal passages, throat and trachea. Besides, the blood supply in your lungs is so well developed that there's virtually no way they can freeze—the circulating blood keeps them nice and toasty.
However, when the air is frigid, it doesn't warm up sufficiently before it reaches the airways leading to your lungs, and so it shocks the surrounding muscles and causes them to contract. This, in turn, causes your bronchial tubes to contract too, which results in that tight, prickly feeling.
Unless you suffer from asthma, breathing in cold air won't hurt you. To get rid of the prickly feeling, just wrap a scarf around your mouth and nose, and you're good to go!
"Perils of Cold Air" (New York Times)
"The Wild File" (Outside Magazine)
"Arctic Mythology" (Alaska Science Forum)