First I tried everything I could to fix it. Reformat the computer, all sorts of stuff.
I called Apple and I got put through to technical support, they replaced my logic board.
Still had the same issue. Spoke with technical support, jumped through 3-4 hoops trying stuff, no dice. Sent them the links (and each subsequent person I dealt with) to the Apple Support Forums that were lit up over this issue.
Upgraded to Lion, still had the issue.
They put me through to Applecare. Rep had me try all sorts of stuff, even moving my monitor out of my office in case it was environmental. Nothing could resolve it. They contacted the engineers, they said to just wait.
This is where I took it to the next level. I said that it had been 7 months since the issue arose, I have had a defective monitor for 3 months and it has affected my work, I thought that Apple's products just worked. Then I asked for a time that I could call back and get my monitor replaced. They couldn't answer, they said just to wait. I asked him if that was the bottom line, they said yes. I was fuming!
I sat on it a couple of days, then called Customer Relations to complain, the first guy said the same thing, just wait. I said that this was unacceptable and I wanted to escalate this. They put me through to a supervisor, who dealt with me for about a week, talking each day.
I sent videos of my issue, sent link after link of Geniuses replacing units, customers getting new units, explained the things that I have gone through, which I shouldn't have to as a customer, that it wasn't a one off issue based off the Support Forum threads. The guy was great, he contacted the Engineers, did his own research, and came back and said they were willing to replace my machine. I almost cried, because it had been such a long battle.
I was always polite, but very firm and made sure they understood how unacceptable it was to have no deadline in sight, when I paid $900 for a monitor, and $1500 for a new computer.
Be thorough but firm, always ask to escalate it and see where it goes. Have video ready to send, threads and research ready to send, give them the amount of time you have suffered through it.
Good luck!