Hey PEU,
Your right on this one, PC AT/ATX power supply is cheap and commonly available. I've a 250watts AT(easier to operate) power supply with all the cables removed. At the back, I added a 240volts illuminated rocker switch which switches the power supply on and off.
First of all, the power supply is hazardous so be careful when working with it. Discharge the large capacitors, using some mains light bulbs to avoid unnecessary electric shock. Most of the capacitors in the power supply should discharge by themselves but please take precaution.
There's a lot of free space inside the power supply but make sure that you do not interfere with the existing circuit and components when adding your variable LMxxx circuits. I use two LM350 voltage regulators for dual voltage output. It has a continuous output current of 3amps. If you want to go for higher current output, use the LM338. It has 5amps continuous current and 12amps peak. My 250watt power supply can handle 8amps at 12volts so it should be fine. The side of the power supply casing is hacked with a window for a mounting a massive heat sink for the regulators /ubbthreads/images/graemlins/smile.gif. On the front of the power supply mounts the potentiometers and banana binding posts. You can even go funky with leds and a voltage meter display would be handy.
The limitation of this setup is that I can only get around 9volts max because of the voltage drop in the regulator. If I need 12volts then I have to bypass the voltage regulator circuit. RussH mention about replacing the fixed sense resistor with a variable one could be a good solution but you have to know the limitation of your power supply. Some ATX power supply requires some minimum load to work properly, and AT power supply is easier to switch on. Use plenty of fuses for your regulator input or the output, they are cheap so no excuses, better be safe than sorry. PC power supply is not designed for this job so keep that in mind.
Aside the limitations this is the ultimate and cheap way of building a variable voltage power supply with high current output.