Microfiber does not work well for me.
For years, I have used Listerine for cleaning parts of me and other things.
It contains a lot of alcohol. I use a cheap big box substitute with the same ingredients.

Here's what I do:
  • 2 or 3 cotton balls
  • dip one in Listerine.
  • Squeeze it out and rub it gently onto the screen.
  • It will bubble up on the surface.
  • Use a dry ball to wipe it off.
  • Initially you will see very little improvement.
  • Use the dry ball (or a 3rd one) to rub briskly (but lightly) on the screen.
  • The little bit of heat produced by friction removes the residue and leaves a clear screen.

Test it on a small section to convince yourself.