Earthing and grounding are different things. Grounding references a circuit to the consensus 0V (Earth) whereas Earthing metal is a safety strategy. Earthing requires a high current, low impedance connection which will ensure that in the event of a fault on the mains side, a fuse or circuit breaker will rapidly open. Over the whole installation, there are standards for "earth loop impedance". Each device which is earthed should itself have a sturdy earth connection to ensure that its path to Earth is adequate.
The computer motherboard has grounding connections, not Earth connections. Any significant fault current passing through it will rapidly destroy it, and likely put high voltages on the exposed parts of the circuits such as connectors (e.g. little 5v USB connections, etc).
So grounding is a matter of low voltage circuit function. Earthing is a matter of high voltage supply safety. The computer motherboards and circuit should be isolated from mains except for grounds. The PSU will have a transformer inside which provides isolation. If you look at the PSU PCB you will (or should) see physical gaps in the PCB bridged only by transformers and optoisolators, to ensure that mains finds it hard to jump the gap and become dangerous to the user.
The Earth connection comes into the PSU. It is robustly connected to the PSU case. The PSU case is then earthed by being bolted onto the chassis, providing chassis Earth. This is the only connection you need to be concerned, for safety reasons, being connected to the chassis. Make sure you have a good, low impedance connection from the PSU to the chassis plates. You don't need to "earth" the motherboard at all. It'll run just fine sitting outside a case on an insulating desk or a towel etc. Bear in mind that much electronic equipment is double insulated, meaning there are no connections from low voltage ground (0V) to Earth at all. But ground connection helps reduce stray interference, so it may be beneficial to ground some of those screw mountings anyway. But they aren't there to provide Earthing.
 Technically at least here in Britain, "high voltage" is only over 1000V, anything below that is "low voltage". I am using the vernacular usage in which anything dangerous is termed high voltage.