HotSpring has introduced a new filter, the Tri-X. They can go into the dishwasher. They are about twice the size of our old filters, but will retrofit - so any of you out there who own HotSpring tubs might want to check them out. Yes, they are more expensive, but they should also last longer.
As to putting filters into the dishwasher: I figure that watever is left after the filters have run through a complete cycle would be minimal - and it's most likely not any worse than the bits of left-over food and grease on the incoming dishes. Plus, the dishes are washed in strong, sudsy detergent, so I don't really worry about getting stuff on them from the filters.
No that I think of it, I wonder if I shouldn't be worried about getting gunk on the filters from the last load of dishes. Hmm..
