And an optician checks in!
OK, I have antireflective coating on all my glasses, which I do not get for free (but the frame reps are happy to give me crazy discounts off the wholesale price). I clean them usually at work with the nice-smelling lens cleaner we buy and a cloth diaper. When I’m at home, and I AM NOT SAYING YOU SHOULD DO THIS, I use dishsoap (Dawn) and a pair of beat up old blue jeans. Yes, that’s right, I use freaking denim. After 2 boys outgrow a pair of jeans, the cotton is pretty darn soft. I will say that I’ve never scratched a lens with my denim method, but once totally destroyed a pair with one swipe of a cheap paper towel.
As far as coatings being the reason not to use your shirt…that’s not why. You shouldn’t use your shirt because you walk around all day picking up little bits of airborne crud, which you are then grinding into your lenses. Not to mention that the oils of your skin will only make your lenses dirtier.
Here’s my dispense talk: “When cleaning your lenses, wet them first and wipe them with a clean soft cloth. Cotton is best. Don’t use paper towels, tissues, napkins, toilet paper or the shirt you’re wearing.” I also inform that fabric softener is waxy and any fabric washed or dried with it will not clean your lenses, but will make them look worse.
Tip on Anti-Reflective lenses: wet both sides of both lenses with either soapy water or eyeglass cleaner made for AR. Gently “smear” the lenses with your fingers on both sides (in a sort of pinching posture with your fingers). Turn on the water in the nearest sink, fast enough to flow well, but slow enough to not be aerated. Slowly pass your glasses through the running water, and most of the cleaner/soapy water will “slide” right off, and the lenses will be nearly clean, needing only a cursory swipe with a cloth. AR coatings are pretty “hydrophobic”, and the more expensive coatings actually do seem to clean more easily and scratch less (and yes, I’ve got cheap AR and expensive AR, and I actually do prefer the expensive stuff. However, my lenses have never ever once been free, and with as often as I like to change my frame, I get the cheaper stuff).
Any other questions? I’ll pop in from time to time 
c