What is important in this whole process is not that the items being worked on are at absolute ground, but that they are at zero potential (voltage) in relation to each other.
You can work on computer equipment static electricity safely by establishing a ground plane. To do this, get a work bench that has enough surface area, and get an anti-static mat that will cover it.
Attach your wrist strap to the antistatic mat so you are now at the same charge everything else is at on the mat.
Set the computer on the mat and jumper the computer case to the mat for a sure bleed-off of static electricity.
Lay all the antistatic bagged or packaged components on the mat. This will bleed off the difference in charge between all the bits and pieces.
Now you can remove the components from their antistatic packaging and R&R as necessary, knowing they will be at zero potential (voltage) to each other. As an added precaution, handle all boards by their edges, you don't need acidic, greasy fingerprints on any of the connectors, its bad for other reasons than electrostatics.
If it is really important that the ground plane be at earth potential, you can get special power cords that only have the third prong (US power) circuit. They are green color coded so you know they are not a regular power cord. Plug this into the computer power supply and the case will be at absolute earth ground potential and through its jumper, will also ground out the antistatic mat.
The above procedures gives you a mobile anti-static work area that can be assembled anywhere to safely work on static sensitive components.
For field work, where this isn't available, unplug the computer, open the case and clip your antistatic wristband to the computer case so you're the same potential.
Touch the antistatic packaging of each component to bare metal on the case before opening to be sure they're the same potential. Theoretically you handling the bag should be sufficient, but this makes sure.
Handle the component by the edges, keep acidy, oily, sweaty fingers off of contact surfaces for corrosion purposes and install them in the system.
In the early days when things were really sensitive to static, all of this was mandatory work practice. Now, most circuitry contained on boards has some anti-static protection built in. It's a good idea to not to challenge Murphy. The first static zap will probably not destroy a device, nor maybe the second one, but it can weaken components to cause future mysterious failures, something you don't need.
Following these practices, I've yet to zap something into oblivion.
Bit of discussion on how anti-static wrist straps work. They're there to protect the silicon from you.