Turn off the monitor. If the screen is dark, it will be easier to see the smears and smudges.
Remember the screen is NOT glass. So use a soft, dry, microfiber type cloth and very gently wipe the screen. The sort of cloth used to clean spectacles.
If the dry cloth doesn't completely remove the dirt or finger grease marks, DON'T press harder in an attempt to scrub it off. Pushing directly on the LCD screen can often cause pixels to burn out.
If necessary, dampen the cloth with distilled water or with an equal ratio of distilled water to white vinegar.