What's the best way to pay off a $20,000 credit card debt (multiple
accounts), given these facts:

   1.  until recently I've been barely able to keep up with regular
        minimum payments
   2.  my credit rating is poor
   3.  I'm now able to pay off the entire debt at once.
   4. I want to improve my credit score as quickly as possible.
   
I've heard that a drastic, immediate pay-off might not be as good for
the credit score as a gradual paydown, but is that really true? If so,
why? If not, why?
And how do you really know?

What's the cleverest way to handle this?