We always had this problem. We like our tea made with a Yorkshire tea bag in a mug and brewed long and strong. But our mugs never came out of the d/w clean, even on a 60 degree wash programme.
So ... (are you gripped yet?) ... after reading threads on MN assuring us that the Eco programme costs less in terms of heat and electricity, even though it runs longer, we have switched to the Eco wash and honestly everything comes out cleaner AND drier. The tea mugs are white again, it's so satisfying.
Give it a go!