For wintertime exercise in cold climates, most people opt to stay indoors - or simply skip the workout. I, however, love exercising outdoors, and recommend it to others. I go out in any weather, fair or foul. I think there is benefit to getting out and breathing real air, and seeing the world that surrounds you.
But, as Michael mentioned on the Life-Cise Facebook page, a lot of us become more sensitive to temperature extremes during and after cancer treatment. Our newly bald heads, our tingly hands and feet need a little extra care during the winter. Extra layers, warmer hats and mittens.
And our newly bald heads and any body parts which have had radiation need extra care during warm months. Sunscreen! Of course, everyone should protect themselves from the sun, but radiated skin in particular needs extra care. You've already had one type of cancer, you don't want to put yourself at an even higher risk of skin cancer. (Make sure you know the long-term affects/risks of any treatments you're getting.)
But right now, much of the country is in the deep freeze. That doesn't mean you need to stay in; you just need to take the proper precautions. Even now, after more than a decade of survivorship, my hands and feet are much more sensitive to cold than they were before. Therefore, I always protect my hands. I wear warm gloves. When I climbed Mt. Elbrus last July, I wore my big expedition-weight mittens. They were heavier-duty mitts than most people wore, but I know I need to protect my hands from the cold. I also spent extra money for warmer mountaineering boots when I needed new ones. When I run, I always wear gloves - even in moderate weather.
I still recommend going outside; I encourage clients to get out as much as possible. Just be sure you prepare. Dress in extra layers. They'll keep you warm, but also give you the flexibility to take off something if you start to get too warm. Don't forget warm clothing for your head, hands and feet - they're likely to be the most cold-sensitive. Be sensible, too. If you're getting chilled, turn around and go home, even if you're not finished with your walk/run/bike. And know your limits. Sometimes you might just need to stay indoors. It doesn't have to be every day, but it might be prudent once in a while.
If you do brave the elements, I hope you will be amply rewarded. On recent cold runs and hikes I have been surprised by a Barred owl landing right next to me, and found some wonder treasure in the frozen creeks. In spite of the 14 degrees of cold today, I will still go out. I will dress appropriately, and I will see what new treasures await.