Smoked,
Be just a tad more specific about which portions of the door panels you need to clean.
Based on your screen-name my assumption is that you've got a '93.
That being the case, are you needing to clean the hard plastic portions of the door panels or is it the fuzzy elbow cushion?
As for the speaker covers, that's not cloth. It's a plastic grate and is removable from the back side of the panel, if you're careful. With the removal of the panel and removal of the grate you can scrub the grate with soapy water and a stiff brush. If you don't want to remove the panel, use a microfiber cloth moistened with a plastic/vinyl cleaner like Vinylex. Don't get it wet so you get runs into the speaker, just keep it moist.
Actually, that microfiber/Vinylex thing works really good on the hard plactic door panel parts too.