Instead of soaking up sweat (like thicker, cotton shirts), quick dry shirts are made from a soft polyester or nylon blend that wicks sweat away from your skin and effectively evaporates the moisture.
Besides smooth woven fabric, you also have the option of wearing more casual knit shirts. These can be polos (in short or long sleeves), henleys, quarter zips and turtlenecks. All of these look fine ...