Which natural (non-synthetic) fabrics are good for hiking winter weather? I know cotton will generally trap moisture and invite hypothermia, while wool is warm but heavy. What other natural fabrics will work well in winter conditions?
There are many excellent natural fabrics for winter hiking:
When I'm enjoying nature I like to wear what's natural.
You probably want wool. Wool has a fairly good warmth to weight ratio, and keeps most of its warmth when its wet. You can get wool products for both base layers and insulating middle layers. "Merino wool" is the style of wool that seems to be popular for high end outdoor wear now.
They can be expensive, but the following companies make very high quality wool pieces for outdoor wear: Ibex, Smart Wool, Icebreaker. More general outdoor companies like Patagonia, Arc'Teryx, and Outdoor Research also offer some wool pieces. Unfortunatly, they're very expensive. Like a lot of "natural" products, wool outdoor pieces are priced as luxury versions of the synthetic ones.
Down may also be appropriate, but I've never seen down jackets that weren't also made with nylon (I don't understand your motivations for wanting natural materials, so I don't know if this is a problem). Down has the best warmth-to-weight ratio of almost any material, but keep in mind that its worthless when it gets wet.
For what its worth, wool is relatively warm when its wet, but that isn't a substitute for keeping dry - if it might be raining where you're going, you should bring some kind of dry outer shell. Unfortunately I'm not aware of any commercially available natural materials for that (i.e., there isn't a natural alternative to a Gor-Tex jacket). Even with nice wool layers, if you get rained on enough, you would still get hypothermic.
As others have said, wool is the best of the easily available natural materials. It's main advantages are:
On the downside:
Down is a very warm natural material and is highly compressible, but has serious downsides that it looses most of its insulating property when wet, and it really soaks up water and not only stops insulating but gets very heavy. For most people that go places often enough where getting rained on is a common consideration, down is essentially useless for clothing. Down can be good for something like a sleeping bag that you have means to keep out of the rain.
There are other natural materials with some interesting properties, but after wool and down they get exotic, and either impossible to find or very expensive if you do. For example, wolf fur is the best for repelling snow, and seal skin is particularly good for waterproof garments.
I know you asked about natural materials, but without explaining why your question is limited that seems silly. Modern technology has not only produced materials that are very good for outdoor use, but has also made them accessible and affordable. For example, polypropelene is even better at insulating when wet than wool, and feels much more comfortable against your skin. A tight nylon mesh is still breathable but much better as a wind barrier for the weight than any natural material I know.
My winter outfit uses a combination of materials. Cotton is no good when wet, but light and comfortable when dry. When you know you can stay dry (which includes managing sweat), cotton can be nice, but be prepared to change quickly when conditions change. For serious conditions I like a polypro "sweater" on first, a wool sweater over that, and a wind shell over that. The wind shell can be a ordinary windbreaker (preferably without extra insulting lining, just a real windbreaker) or a rain shell, depending on conditions. You can mix and match these layers to suite a variety of conditions.
In winter particularly, be aggressive about changing your layers for the conditions. You're going to get hot hiking uphill. Strip down as appropriate as soon as that happens. As soon as you get to the top, put on a layer or two before you cool down. It's a lot more comfortable to switch layers when you're warm.