I would recommend a thin microfiber cloth considerable texture of it. Glasses cleaning cloths are generally too small and too smooth, so they usually just end up changing on the screen than it is cleaned. I have found that often be the case with many screen-cleaning kits. They usually give you alcohol-based spray that helps reduce this so they can sell you refills, but you should not need anything more than water.

Similarly, if you have a long pile microfiber cloth, they tend to just accumulate dirt / grease time and make things worse.
MPJ Microfiber cloth made for cleaning glass, screen and lenses is pretty good.

Do not use paper towels / tissues to clean the screen. Paper-based products are abrasive and may damage the surface of the screen.