DIAGNOSIS You're right to be careful. The screen has a strong matte anti-glare coating that can look speckled or "dirty" even when clean, and harsh cleaners can damage it.
FIX Power off the monitor so you can see streaks. Use a dry, soft microfiber cloth (like the kind for glasses) first to wipe away dust with light pressure. If you need moisture, dampen a second microfiber cloth with distilled water only—never spray liquid directly on the screen. Wipe gently in one direction. For stubborn spots, use a 50/50 mix of distilled water and white vinegar on the cloth, not the screen. The vinegar helps with grime without harming the coating.
IF THAT DIDN'T WORK If there are persistent streaks or what looks like oily marks, it might be residue from someone using a bad cleaner. Try a monitor-specific cleaning solution that explicitly says it's safe for anti-glare/matte coatings. Apply a tiny amount to your microfiber cloth, not the panel.
HEADS UP Never use window cleaners, Windex, paper towels, napkins, or anything containing alcohol, ammonia, or abrasives. These will permanently haze, strip, or scratch the anti-glare coating, leaving a smeared, blotchy look that you can't fix.