This has been something that intrigued me for a long time. I had memories of frozen items coming in from the washing line, and, after ironing, were 'aired' on a rack or radiator. So I have been experimenting with the idea.
The first time was a set of sheets that I put outside. It was cloudy, but alas the partly sunny forecast from the weatherman was a snow shower by midday!  I brought the almost frozen sheets inside and put them on the indoor line to dry. They smelt great but not the best result.
Last week we had brutally cold weather courtesy a cold front from the north. The cold nights in part were from the snow cover, but also because there was no cloud cover. The sun was shining, the humidity was low and I had shirts to dry.
You feel a bit silly putting on boots and trotting over snow to the line with a basket of damp laundry, but I did it anyway and hung the shirts to dry.
Over about 4 hours (11 - 3pm) they dried - just. The collars were still damp, but after a quick iron, they were hung in the laundry room overnight and were ready to be put away by morning.
So clearly the key is a very dry clear day, and a good number of hours, but drying linen on the line outside is very feasible even when there is snow on the ground!