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twicx
So from what I can tell, the plan is to take 3 x 12V wires and 3 x GND wires from the ATX and "group" them so that they all connect to the HB + and GND respectively.
Then take a single +12V and a single GND from the ATX and go to the board power (for all the other stuff)
and then one last +12v and GND for the fans.
If you do that, your heated bed and fans will always be on when the power supply is on. You'll have no control over it. You don't mention what electronics you use, but typically ALL the power wires go to the electronics (RAMPS, Melzi, Gen7, etc) and then you have wires that go to the heated bed, fans, hot end, etc. Alternatively, some setups will hook the heated bed more directly to the power supply but use a a solid state relay to do the switching of power instead of the board electronics.
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With the other cables in the ATX, I've the ATX case open, and they will all be cut short, and heatshrink put on the ends of them so that they'll be neatly bundled off INSIDE the ATX case.
You'll likely also need the green PS_ON wire to turn on and off the power supply via software. Some power supplies won't put out sufficient power on the +12V rail unless you have a load (typically a 4.7 ohm 10w resistor) on the 5v rail too so before you go snipping wires, double check that. You won't know until you try if you need the dummy load. You may also want to keep a +5v line around unclipped as well. You might need that down the road if you want to add a servo and the arduino can't supply enough of it's own +5V juice. The rest you can snip.