My dash has a built-in tray on the driver's side over the instrument cluster with a cup holder that's too small to hold any cup I own, an openable compartment in the center over the radio that's just the right size to hold a clipboard or my copy of Next Exit, and another built-in tray on the passenger size with another miniscule cup holder inside the tray. There isn't really any place to put a cover. Glare doesn't bother me much anyway; any reflections of the dashboard on the inside of the windshield are usually from things I've stashed in the built-in trays, not from the dashboard itself, and the reflections are low enough in my field of vision to not be terribly distracting.
Felt cloth would be perfectly adequate for your cover, if you can find it, in whatever color best suits you. You've never seen glare off a pool table or casino gaming table, have you? They're all covered in felt.
Engineering: Finding complex solutions to simple problems you didn't even know you had.