The best way to clean your Traveling Culinarist bag is with water, just plan water. You can either rub the dirt off with your hands or else you can scrub the bag with a soft brush. You should be able to get most of the dirt off this way. If you feel that you need to use soap, use a mild or pure soap and rinse the bag thoroughly to remove any residue.
Don’t use a dryer of any sort, just be patient and hang dry or place on a dry surface and wait for it to dry.