Computer inside aluminum plated wooden box
Grounding is a really fun topic among EMI/EMC compliance folks. Here is my take on it for computers:

With so much high-frequency content from the various clocks and data interfaces, coupled with the large number of SMPS power supplies on-board, entombing the system inside a metallic case of some type is virtually the only sane way to pass FCC/CISPR radiated emissions testing. The case protects the hardware and nicely attenuates significant amounts of noise, helping you pass testing.

On the motherboards I've designed, mounting holes and "chassis grounds" were single-pointed to digital GND on my PCB itself, through a high-impedance resistor -- all we wanted to do was bleed off charge and not directly conduct ESD events into our circuitry. Some folks may keep them completely separate, and depend on a metal case to link all those paths together. Our metal casing was tied to AC earth ground -- noise got shunted to AC earth, and if we had a short circuit, the case would not remain at a dangerous potential.

The danger with a metal case that is not grounded to AC ground or similar is that if you have a short-circuit inside the unit (AC or DC), the entire case will now float at that potential. You can imagine this being dangerous if you unknowingly had AC live shorted to metal -- anyone who touched that would be in danger.

In your case (hah), I would do this:


Beep out (measure resistance) from the ATX motherboard mounting holes to a known GND point. If it's 0 ohms (short), you're done and fine, since your GND is already connected.
If it's open, check the holes to each other. If they are separate (open) to each other as well, then you also have nothing to worry about -- those were designed to allow screws to mount the motherboard and do nothing else.
I take it it's a wooden box, with aluminium siding on the outside. Does this aluminum make contact with a metallic EMI shield from the motheboard? Or the ATX PSU? My gut says that metal should be earthed to save you in case any kind of high-voltage shorts to the chassis -- better the device goes "pop" than you.


Here is a related SE question: Is it dangerous if I do not connect the ATX shield to ground? 

This is a potentially (hah x2) divisive question and I've love to have a discussion on it with folks.