[Warhound]

Maybe I should clarifiy what I meant with damp.. Just a few droplets of water in the centre (fe. turn a bottle upside down once while pressing the paper against it) then fold it over twice so you have 1/4th of the normal size and work with that. Use 2 sheets if it's kinda thin or your finger grease will push through the single piece and leave smears aswel.
The water never ever runs down or get's in cracks really..it's just enough to leave a moist shine on the screen and ease the removal of grease etc..If you aren't fast it will even dry up before you get round to buffing it out with the dry paper towel. Another point is to do it before using the monitor so it isn't hot, making the water/product dry in faster.

Definitely NOT a wet cloth like you use to clean windows ,cars or anything like that. Like you say, water and the inside of electronics are a bad combination.

Edit : @Qpassa sounds pretty much like my method ,except our toiletpaper leaves way too many small particles behind so I use paper towels for both stages.