As always, people will disagree with this. Here's how I have clean the viewing screens.
First, there are two kinds. The early screens (usually fixed in the camera) were made of real glass. Later screens are plastic, and those can be scratched very easily.
For the OM (single-digit models), use the little tool to remove the screen. Use compressed air to blow away the debris.
If the screen is grimy, and some are for some reason, clean it by putting some dishwashing liquid on your fingertips. Under running water, hold the screen by the edges with one hand and thoroughly wet the screen on both sides. Use your fingertips with the dishwashing liquid to gently clean both surfaces of the screen. Do not do this is if your fingers are heavily calloused!
Once clean, still holding the screen by the edges, use compressed air to dry the screen on both sides. The water droplets should simply be blown off the edge of the screen.
Reinstall the screen. Before you reinstall the screen, clean the underside of the prism. Be careful not to damage the meter needle (if there is one).
Also remove any particles of old film, but to be truthful, you should re-foam the camera before you do any of this work. Cleaning the viewing screen should be among the last steps.
I've done this with maybe 40 or 50 SLRs and have never had a problem. It's all about handling these items carefully, especially the plastic screens - which can be easily scratched.
The reason this works is that the running water helps to wash away any debris while your fingers spread the dishwashing fluid across the screen and clean it. The running water will then wash away all traces of the dishwashing fluid, leaving you with a very clean screen. If you have a sprayer on your faucet, use that as the final rinse. Just be sure to have a firm grip on the sides of the screen. If you are cleaning a glass screen, don't grip it so firmly that you break the screen.
As others have said, dust on the viewing screen has no effect on the actual photo. Even so, I find those black specks to be annoying and distracting.