I just noticed that adding a tea spoon of bleach to a nearly full cup of water and letting it stand this way for a few minutes totally eradicates all the tea/coffee stains and leaves the cup as startingly white as Ben Affleck's dentures. Moreover, I can transfer that water to repeat the process on 3-4 more cups before the diluted bleach loses effectiveness. No need to scrub anymore, so yay! Only problem was with a plastic cup. Unlike others made from China or Corelware, the plastic seems to have incorporated the bleach smell to some extent and no rinsing with hot water has totally gotten rid of this so far. 
Still, say bye bye to ugly cup stains.
Still, say bye bye to ugly cup stains.