DIAGNOSIS: You're right to be careful. The screen has a matte anti-glare coating that can be damaged by harsh chemicals or abrasive cloths. Water alone often just smears grease around.
FIX: Here's the safe, shop-teacher method. 1) Get a soft, clean, lint-free microfiber cloth—the kind you'd use for glasses or a camera lens. 2) Use dry, gentle pressure in a circular motion for light dust. 3) For fingerprints or smudges, dampen a second microfiber cloth with distilled water (tap water can leave mineral spots). Wring it out completely so it's only barely damp. 4) Wipe the screen gently, then immediately follow up with the dry side of your first cloth to buff it streak-free.
IF THAT DIDN'T WORK: For stubborn spots, you can use a 50/50 mix of distilled white vinegar and distilled water. Same process: a barely-damp cloth, then immediate dry buffing. The mild acidity can cut through residue. Avoid the edges where liquid could seep in.
HEADS UP: Never spray liquid directly onto the screen. Never use window cleaner, isopropyl alcohol, acetone, or paper towels. These will permanently haze or scratch the anti-glare coating. Always apply liquid to the cloth, not the monitor.