I both do and don't wash the milk grain. Be aware that high calcium water [it leaves white marks on sinks] will kill the milk grain very rapidly. Hard water makes it quite ill. I do wash the grain occassionally - but when I wash it, I then remix it with a few tablespoons of its previous milk mix, I only wash it in a dish of water to release the baby grain that is in its culture. I then mix half and half the previous milk with new milk to raise the tiny baby grain. Water quality is terribly important in my experience, I nearly wiped out my entire culture of milk grain in just two rinses with hard, calcium heavy bore water. Rain water from a clean roof is an excellent option, some town water doesn't seem to agree with the grain either. As a general rule, I only wash the grain once every two weeks when my new baby-grain has come on satisfactorily. I agree with the other poster, the 'sticky' milk on the grain appears to be important to its own overall health - without that mix to start the baby grain with, it simply doesn't grow like its parent culture do - it appears to need that 'sticky' milk to learn the art of culture just like most other babies need adult guidance.
Water grain is even more difficult in my experience when it comes to water and the only time I'd risk 'washing it' is in clear, clean fresh rain water ... I have successfully killed the water grains using tap water and bore water in a single rinse and I only rinse it at all if it has been left a day or more too long between water changes.
I hope that helps!