We had exactly the same problem, so I went to a local farm supplies shop and bought a zip up wash bag that horse riders use to wash their horse blankets.
It's got tiny holes in the page so the fleece gets washed, but the hay (and any other bits) are retained in the bag.
If you can't find one, you could buy some cheap pillow cases, and sew zips into them for a similar effect.
Hope that helps.
Edit: Just had a look on Amazon and they have jumbo ones for 16 quid.