Quoting Gentleman of Fortune
"It doesn't have to be a fancy silk affair either! A simple linen job, a square yard, will work."
and if I recall correctly Kass said linen would be more commonly correct for a neck cloth.
and in the summer, silk is cool enough but linen is cooler and breathes much better...
But you can also check it out with the experts over in Twill, Kass, GoF, Blackjohn and of course last but not least Foxe...just search neck cloth, head scarf, neckerchief, sashes and you will find quite a bit of serious discussion regarding yer topic.