If you reuse the cloth or let your fingers touch stains will reappear. Here's the technique I've used for years:
Brush away dirt with a fine brush.
Clean with Cleaning kit liquid with lint-free cloth (optician's liquid is good)
Use another cloth to wipe dry.
For the 'stains' use a small piece of newspaper and wipe with circular motions. If you don't like the newspaper idea, wear surgeon's gloves instead.
If you come into contact with the monitor directly, do everything again - oil spreads.
This way, when clean, there isn't a single spot visible against black, from all angles. Never touch the monitor directly, always use the sides to move.
Hope this helps.