What is the best way to desolder?

I have tried solder wicks, single bulb solder suckers, pen-like solder
suckers, solder suckers with integrated soldering irons, and I've
tried heating the item and shaking off the solder.

My current best method is heating the solder and blowing it off with a
plastic soda straw. This sometimes leaves solder all over the place,
and I have to use my fingernail or a chopstick to nudge the solder off
places where it is shorting two connectors, or where it might fall and
short something (which is everywhere).

I ask now because I have, that I found in the trash years ago, about
12 Western Electric circuit boards, obsolete now, each with up to 7
6-pole double throw relays. Needless to say, Western Electric used
high quality relays, but when I try to desolder them from the boards,
I lose a lot of the connections (there are 20 of them), and often I
lose one of the two connections that goes to the relay coil, and I've
ruined the relay.

I need a better way to get these relays disconnected from the circuit
boards.

Help?

Meirman
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