Jeff,
I have found the only way to get the bolt out easily is to use a small powerful magnet to lift the bolt straight up. If it is tilted in the slightest it doesn't seem to want to come out. I use one of the strong rare-earth magnets from an old disk drive autopsy - works like a charm.
Once you get it out and finish the cleaning, the real fun starts when you try to get the bolt back in over the bolt handle.
If anybody has any other good tricks-of-the-trade, like you, I'm all ears.