Mold can grow on anything....I would think that it's more likely to grow on hay/straw than wood, but there are a lot of resident organisms growing on tree limbs/bark that can bloom when put int a wet airless environment.I'm hoping a local arborist gets back to me as I do like the look of wood chips too. I did initially have bark but I was worrying about it composting and even going mouldy so I raked it back.
For now they have two hay bales they are enjoying destructing. I have decided to add some stepping stones and put stones under the soil for added drainage. It is covered with a roof drainage systems, we've had such awful weather and it just seemed to have retained the extra water.
Airflow and sunshine is the best deterrent, but...in your case I'd just keep an eye on things and keep them stirred up with a rake or scratch thrown for birds to do the stirring.