Another method: For dust and dirt that is on the surface of the Antron, roll Gaffer's tape (like duct tape, but doesn't leave sticky residue behind) around fingers with adhesive side out and gently pat the material. I've had good luck lifting dirt off with this method. Sometimes even light washing or scrubbing of any kind can grind the dust deeper into the fleece, making a larger mess. It's best to try to clean it without adding liquid first, and then move to the wet methods.
Good luck.