One of the greenest options for napkins, is to used recycled fabric. Textile-production is generally NOT an earth-friendly process, and the production of cloth - while it doesn't contribute to deforestation, DOES have a pretty hefty carbon footprint.
So, one of my favorite napkin materials is used denim. The ones in the photos below came from a pair of torn-up and outgrown jeans belonging to my stepkiddo.
As most people can attest, jeans make an EXCELLENT place to wipe your hands (sigh...). So it stands to reason that they'd make good napkins, too. And when they wear out as napkins, they make excellent cleaning rags.