We all forgot at some point some tea or coffee in a mug. If you continiously do this, you will notice that the stains do not go away with just cleaning them with dishwasher soap. You need one more step in between.
Soda and vinegar is the solution.
Leave a teaspoon of soda and a tablespoon of vinegar sit there for hours and then wash it like normally with the dishwasher soap.
Yannis and Stella @Apollon, Naxos 