Plain cardboard boxes make great weed barriers. Remove any tape and stickers if you can.
Pulp egg cartons are biodegradable and can also be used to start seeds in the spring!
Amazon's new paper padded mailers can be used just like cardboard.
Paper bags that are clean and free of food/oil are great for small spaces. Newspaper works great too!
Stained or torn clothes made from cotton, wool. linen, etc are biodegradable.
Once you arrange your weed barriers, wet everything with a water hose to help it lie flat.