I have never used an antistatic strap, and I have never had any problems. I also don't shuffle my feet about on the carpet while I'm working on a machine... A good way to proceed is to:
Leave the machine plugged in to the mains, but turn it off at the wall socket. That's right - think about it - it is still earthed. This is not strictly necessary, but in electrical terms it is quite safe.
Do NOT work inside the machine unless you are sure that the internal power supply is completely drained... Switch the machine on and off a couple of times to make sure, or just switch it on (N.B. it is still OFF at the wall) and leave it for a while. There could be quite a bit of reserve capacitance in there.
DO touch the metal chassis of the machine before you fiddle about inside it - this will discharge any static on your body to earth (hopefully through the still plugged-in mains lead, but if not connected that way there is still enough metal in a good case to do the job). Do this frequently - if you have modern materials in your clothes they might create static just by your moving about. Simply touching the chassis every now and then will prevent any build-up of charge. If your clothes do create charge in this way, change them for something less fashionable. :-)
I would rather make sure I had earthed myself through touching the case than wear an antistatic strap. If you have to get up and cross the room then you will probably need to take the strap off to do so, and walking on a carpet might create enough charge to damage some components. It is swifter, easier, and more convenient to simply make sure you earth yourself every time that you prod about in the machine.
