I had that problem too. Tried all sorts, bark chips, gravel, straw, boards - nothing worked well enough. Eventually, decided to keep the boards down, remove the sodden straw, level it all off to the height of the boards, lay terram (weed fabric), bought cheapest 20ml gravel and put that on top. Must admit I was given the weed fabric, but the gravel cost £90 for two tons delivered - area is about 36feet by 18feet.
My son drilled holes in the brick wall that runs the length of the run so the excess water can drain away to the burn alongside, and now I can hose it down, spray with Jeyes when they have gone in for the night and it's OK for them in the morning. So much easier to walk on too. Eggs are laid inside in shed anyway so they don't get dirty.