First, let me say that many new polyesters are designed to be breathable and are for the most part. The cheap ones are not, but the better ones are. All polyesters I've tried, even the anti-microbial treated ones, do get stinkier than natural fibers like wool, silk, cotton or hemp though.

But wool! Itchy? Too warm? A thin wool t-shirt is my favorite warm weather bike shirt unless the temperature rises above 80 degrees in which case I move to a polyester mesh. In fall, wool is great and layering a few wool items that can be tossed in a bar bag or pannier can allow for riding in conditions from just above freezing up to 80 or so if you have rain gear to use for a top layer in rain or sleet.

Wool rocks. Merino Wool! Do not use a heavier weight directly on the skin. Shetland for example, is great for a second layer but not on the skin. Wool shirt. Wool sweater. Wool socks. It's the miracle material. Stinks less than anything, Wicks better than cotton. Feels good on the skin.