Diagnosis
You're likely dealing with a combination of dust and skin oils, and a dry cloth will just push the oil around. Using the wrong cleaner can permanently damage the anti-glare coating.
How to Fix It
First, power the TV completely off so you can see smudges. Use a dry, soft microfiber cloth (like for glasses) to gently wipe away any loose dust in a single direction. Never use circular motions. If smudges remain, dampen a second clean microfiber cloth with distilled water only. Wring it out so it's barely damp, not wet. Gently wipe the screen in straight, vertical strokes. Immediately follow with the dry microfiber cloth to buff it dry. That's it.
If That Didn't Work
For stubborn spots, you can make a very mild solution. Mix 50% distilled white vinegar and 50% distilled water. Dampen the cloth with this mix, wring it out extremely well, and wipe gently. Follow immediately with a cloth dampened with only distilled water to rinse, then dry thoroughly. The vinegar helps cut grease.
Heads Up
Never spray liquid directly onto the screen—it can run down into the edges and damage the electronics. Never use window cleaner, alcohol, ammonia, or abrasive cloths (paper towels, old t-shirts). They will strip the coating and leave permanent cloudy patches. Patience and distilled water are your best tools.