I wear ordinary cotton bandannas. I already had dozens of 'em. Some belonged to my granddad from his gardening days.

I have long hair and wore 'em anyway, so they work fine under a helmet. And I have so many I can wear a fresh one every ride and wash the whole batch only every two weeks or once a month. Since I began drinking electrolytes with water this summer my sweat is much saltier and the bandannas get white streaks and crusty after one sweaty ride, so they need to be changed every ride. That's impractical with more expensive caps made of merino wool, poly wicking fabric, microfiber, or those spiffy short billed cycling caps.

Also makes it easier to keep the helmet clean. The Velcro'd pads inside the helmet stay clean longer -- I just wipe down the inside of the helmet after a ride. Only the straps need more thorough cleaning so I wipe 'em after each ride and once a week soak the straps by propping the helmet over a bucket of warm soapy water and letting the straps dangle and soak for 30-60 minutes.

But to keep my ears warm I wear springy microfiber and neoprene ear guards. Fits like headphones, behind the head and under the helmet. The spring form holds the fabric taut and I can hear pretty well through the material, so it muffles some wind noise but I can hear traffic, etc. Worked fine last winter for rides up to an hour in the 20s.