To supplement the excellent advice you've already received...
If you don't mind destroying the connector, then removing the pins one at a time is a viable, if tedious option. However, the impatience often caused by this technique is a recipe for lifting pads or tearing out vias along with a pin which isn't quite as desoldered as you thought. So be careful.
All of the operations below are made easier if you have a bottle of liquid resin or flux. The resin helps to conduct the heat evenly and carry it to the solder. It also removes oxygen from the solder so that the solder does not oxidize so much.
To make this work, first mechanically separate the pins. Sometimes you can just slide the plastic housing off of the pins--if you're lucky. Then, I recommend not using any desoldering braid at this point. If you reduce the amount of solder in a hole, it makes it harder to heat the pin evenly (the solder carries and spreads the heat). Before you start using any heat is a good time to apply solder flux to all the pins. Heat each pin and gently extract it with a tweezers or small pliers. Do not twist any pin, nor tug on the pin. The pin will extract easily when the solder is properly melted. If you must use any force at all, you are in danger of pulling the via out of the hole (the copper lining the hole).
After you have all the pins extracted, then use braid to clean the solder out of the holes. During similar operations I've sometimes found it very difficult to get teh solder to exit the hole. A sewing pin on one side and a soldering pencil on the other side of a hole is an effective way to poke the solder out of the hole. Gently insert the pencil's tip into the hole, melting the solder on your way in. Stick the pin in the other side until it contacts the pencil. Now move the assembly so that the pins sticks through the hole and the pencil touches the pin, keeping the solder melted. Withdraw the pencil. Let the solder solidify (a few seconds), then withdraw the pin. Now you have a little volcano cone of solder sticking out of the hole. With a little practice you can do six or seven holes per minute.
Considering the tedium of the above repeated 120 times, a heat gun should be sounding pretty good about now, yes? Doing some quick searching, I don't see anything like what I bought. I have a Milwaukee heat gun adjustable from 100F to 1000F with a continuous dial. It cost me $40 at Home Depot about ten years ago. The closest thing I found was this for $100:
http://www.internationaltool.com/Heat-Guns/Milwaukee-897720.aspx?utm_source=yamar09&utm_medium=pdc&utm_campaign=other
Maybe you can find one for less. If you're going to do this much, it's well worth the investment even at $100.
If there are any other components on the board you'll be working on, cover them in modeling clay. You can literally blow components away with a heat gun, by melting the solder and having the heat carry them away. Modeling clay is an easy way to prevent this. Just roll some little snakes of clay and press them over endangered components.
Apple liberal soldering resin to the pin/solder joints. Figure out how you're going to support/hold the board when it reaches 800F. Sometimes I use a pliers to grip an appropriate part of the board. Sometimes I build pedestals out of modeling clay and use those. Arrange things so that when the solder loosens, the connector can fall down out of the board. Gravity should be your ally.
Set the heat gun to 800 - 900F. Set a dental pick or similar pointy implement nearby. Apply heat to the region of the circuit board to be desoldered. If you like, start a stop watch. It will take about 2 minutes before the solder actually starts to melt, most likely. Use the dental pick to poke the solder around the pins. At first it will all be solid. Eventually, it will be soft and the pick will penetrate it. Make sure the solder of all the pins is liquid. If any regions are still solid, that area needs more heat gun attention.
When everything is suffiently melted you will be able to press the pins out with the pick or a small screw driver without using any force to speak of. Pressing a few pins on each side of the connector will cause the whole thing to fall out of the board.
Turn off the heat gun. Set it somewhere safe. Clean the solder out of the holes on the circuit board. Remember, everything in the area on your bench is *HOT*!. I keep a 12" X 18" X 1/16" sheet of steel on my bench as a base for this kind of operation--plus it's a good anti-static mat.
Do not try to combine the two methods. That is, don't try to remove all the pins at once using a soldering pencil or gun. It *might* be doable with chip quik. If you put a big ole' bead of the stuff between all the pins of the connector and poured heat into it through a soldering gun or two, you *might* be able to get all the pins melted at the same time. My experience is that this method isn't good much beyond two linear inches of pins around the circumference of a chip.
