Simple solutions:

Natural fibers. Certified organic fibers or at least naturally grown, as well as processed without out heavy chemicals and pesticides. There is room for plastics on our planet but not in a volume we creating it right now without workable way of decomposing it. Hemp and organic cotton grocery bag will last for a long time, since hemp is one of the  strongest fibers and sustainable plants. But when it finally falls apart (hopefully after many years of use) you can throw it into compost and improve the soil. Help promote the message to Live Green!