What's so wrong about keeping your cups upside down? Whenever I wash the dishes I always place my cups upside down on the shelf.
Ditto. I have one of those non-slip shelf liner thingies and place the cups upside down on them. I surely picked this up from my parents, however the logic is sound. It prevents things from getting inside and collecting in the cup. Therefore you can just take the cup out and fill it up without fear of what has gotten inside it whilst in the cabinet.