Our dishwasher would do this too. It wouldn't collect in the glasses. The gritty sand you described would stick to all of our dishes.You can run an empty cycle and fill your detergent dispenser with lime away. I would use the gel lime away. It is safe to use in your dishwasher. You can also throw a handful of citric acid in the bottom too. I would use both together. You can buy citric acid at a health food store or you can use Tang. I would run this cycle whenever I noticed the grit coming back.
You might also want to check the flap on the hose leaving your dishwasher. If it is old and doesn't close the particles with flush back into your dishwasher.
Good luck! Our small town repairman told me I needed a new dishwasher or at least a new motor and I used the dishwasher for 5 more years. We really have hard water here. When I bought my new dishwasher last year, the sales lady told me to place a bowl of vinegar in the bottom of my dishwasher once a month to keep the dishwasher clean from the hard water.