I have had to go through this with all 3 of my kids, ages 11, 6, & 4. I know first hand how frustrating it can be. I try to stay very consistent with my response. When one of them would start to whine, I would turn to them and calmly state that I don't listen to anyone who is whining. Then I would tell them that when they are ready to talk to me without whining, I will listen. Then I would turn and walk away. This has worked with whining, crying, yelling and screaming.
With whatever you decide to do, the key is to be consistent. This is one of those things that will not go away quickly. It will most likely take a long time. All of my kids revert back from time to time to see if they can get away with it.