I am blessed/cursed with a black walnut tree that drops pods all August. Any tips for getting ugly brown stains out of plastic lawn furnitur

If the tree is yours, you might want to consider having it removed because of the toxicity of black walnut roots to some other plants. Tree removal can be expensive, but black walnuts are very messy trees. You should be able to sell the wood.

I think you can make a paste of trisodium phosphate and bleach to remove mildew stains. Might work for walnut stains.

My other suggestion is Goo Gone which is made with orange oil.

I'm willing to try! Barkeepers friend, dish soap and Dow scrubbing bubbles doesn't seem to do much. Thanks!

Have you tried rubbing out the stains with some cooking oil?