Let me add my two cents, same as harrymc's answer, based on no scientific method except 20+ years of empiricism.
I've been vacuuming computers for decades, I've not killed one yet.
I use a portable, battery-powered vac [Dyson] thoroughly sprayed with anti-static household polish - Mr Sheen [if that doesn't exist where you live, look for the one that used to be advertised as not making dust stick to your big old CRT TV screen in the 80s… or ask your folks;)
You can help this out in tough to reach places with a cheap household paint brush.
As I live in a territory where earth connections are compulsory, I leave the computer plugged in, but the socket switched off. This keeps the case earthed at all times. I keep myself in contact with the case as much as possible whilst cleaning, though I've never bothered with a wrist strap.
I've never taken any precautions other than these; I don't avoid the motherboard or anything else, it all gets the same treatment wherever I can reach.
Since last year, I've actually switched to a mains-powered blower - always wanted one, never got round to it. I bought an expensive, filtered, high-power one, nearly breaks your wrist as it starts up. I also have sprayed it with the anti-stat polish. The Dyson gets used 2 or 3 times a week round the house; the blower will get used twice a year, so it's a bit of a luxury item I suppose. The clear advantage is it can clean bits no vacuum can get to - rapidly.
I haven't yet managed to damage a fan by spinning it up too fast - though I did manage to break one by revving it up then touching it with the blower nozzle… my own stupid fault, but be warned;)
Also - do it by an open window, if not actually outdoors - or you'll be vacuuming the whole room for an hour afterwards.
In all, the blower does in 2 minutes a better job than the vac in 30. I'd go with the blower any day, but it's a $£€100 expenditure that doesn't get used for much else.
General note on 'grounding'.
The US calls signal ground & electrical earth by the same name, which can lead to some confusion. The UK separates the terms. 'Ground' is a signal ground - e.g.the extra stripe on your headphone jack, to give the signal a 'return path'.
Earth is mains electrical earth, the green/yellow wire that leads directly to an actual spike into the ground, either in your cellar or back at the nearest electrical sub-station.
In countries where earth is compulsory, your safest way to properly earth your computer [or anything] whilst stripping for maintenance is to leave its mains cable plugged into the wall, with the power switch off. This leaves the earth connected, but disconnects the power safely. In this way, touching the casing sets everything to a known equilibrium.
If you live in a very dry area, or ever get static spikes from things around the house in general, wait for a rainy day.
Don't wear plastic shoes & run up & down a nylon carpet. Don't rub the cat with a balloon or amber rod - this should, of course, go without saying;)