Generally speaking, it's the synthetic fiber fabrics that have proven to be advantageous when it comes to workout underwear.  That being said, there are some natural fiber fabric choices that are breathable and moisture wicking. 

Nylon: Nylon is the most popular synthetic fiber for running and athletic apparel, nylon often originates from a "fossil fuel" like coal or petroleum.  Nylon is more recently drawing a lot of competing natural fiber materials which can be more "eco-conscious."

Polyester: Polyester is also a "fossil fuel" / plastic-based fabric. Polyester is lightweight, durable, and breathable and stays much drier than a lot of other natural fabrics. Unless added chemicals are used, polyester doesn’t have the antibacterial qualities of some natural fiber fabrics.

Spandex: This synthetic fiber is almost always mixed into either another synthetic fiber or cotton.  Spandex is valued for its ability to stretch and expand.  The stretching and expanding characteristics offers athletes unrestricted movement capability, while the fabric retains its shape over time.

Bamboo: Bamboo is a great soft, sustainable, eco-friendly natural fiber. It’s naturally sweat-wicking and anti-bacterial.

Wool: Although not that common, merino wool is well-known for its ability to regulate temperature in both hot and cold weather.   Beside being temperature regulating, wool is naturally breathable, sweat-wicking, and anti-bacterial.

Modal: Modal fabric is a naturally sourced fabric fiber made from beech tree cellulose . As such, modal is generally considered an "eco-friendly" alternative to cotton or synthetic fibers.

Choose your workout underwear fabric carefully.  That does not mean you necessarily have to break the bank buying an expensive pair of workout-specific tights to avoid the bunching that boxers may cause, or the painful chafing that those damn cotton briefs bring with them. You can choose a pair of workout underwear specifically designed with your manhood in mind,  including anti-odor, stretchy fabrics, and contoured pouches up front that are made to perfectly cup your junk.