Cowboys and WW I fighter pilots wore them to keep their wool shirts or shearling lined jackets from rubbing their necks raw.

In my younger days I had a job driving a tractor in the winter. I had an Army surplus wool overcoat that was great for staying warm, but it would eat my neck up when I would turn my head. Solution was to wear a bandana.

Silk is excellent. It is very warm for the weight (but expensive).

I have some 34" X 34" cotton bandanas that I use similar to these:
http://www.sportsmansguide.com/net/cb/cb.aspx?a=217660
These are larger than a regular hankerchef or bandana.

I have one in my day pack or pocket whenever I am outdoors. In cold weather they help keep your neck warm and keep the wind from blowing down your shirt. In hot weather they serve as a hand towel, sweat rag or just rag. If you don't have a hat, they can be used as do-rag fashion or as a sweat band.