desoldering is the real art, isn’t it? At work I have a powered desoldering iron. At home I recommend either using a sold-a-pullt with your regular iron, or else a desoldering iron with a rubber vacuum bulb.
http://www.elexp.com/sdr_0822.htm
http://www.elexp.com/sdr_0849.htm
both are pretty cheap. I find braid only good for smt or non-plated holes.
Usually at home I use the bulb iron. Start with a fully soldered joint. If you have previous unsuccessful attempts to desolder, fully fill the holes with solder again.
Before applying the iron to the board, squeeze the bulb fully. Then put the iron on the solder joint. In a second or two the solder will melt.
Do not press down hard against the board or else you might lift the pad. Move the iron back and forth or in a circular motion as you quickly release the bulb. Do not stop moving the iron! You want the lead to be moving while you suck the solder out and immediately remove the iron. If there is still a little solder left in the hole, the lead movement will help prevent the solder from bonding to the component lead.
Now put your finger on the component (in your case the resistor network) and wiggle it a little while you look at the lead on the other side of the board. Does the lead wiggle? If so, good job! If not, wait for the hole to cool and lightly prod the lead with your fingeernail or blunt instrument. Hopefully the lead will work free easily. Do not apply force or you can damage the board.
Maybe you can’t work the pin free. Don’t worry. Move on to the next pin, and try again, starting from a fully soldered joint.
Keep doing this until you’ve finished every pin. Ideally the component will just fall out, but maybe you have 1 or two pins that are still sticking. You can try to apply heat to one hole while gently wiggling the component. Keep wiggling it even after you’ve removed the heat. When the trace amounts of solder in the hole harden back up, they will most likely not bond to the pin.
I’m pretty experienced with this, so I’m confident to do it without damaging a board. In the past I’ve ruined my share, so if you’re unsure, you’re better off cutting the component off and replacing it.
You want to avoid wiggling or moving the solder joint when it’s hot but the heat has not transferred to the solder. In that case the only thing that will have any flex will be the adhesive that bonds the copper to the board, and that’s how you lift pads and traces.
Good luck.