Combine a few drops of a mild cleansing detergent in a bucket filled with warm water. Pour one-half cup white distilled vinegar in the bucket and swish it around with a soft cloth.
Dip the cloth completely in the mixture and wring it out. Wipe down the shoes thoroughly to remove a surface layer of dust.
Dip a scrub brush in the mixture and shake it out. Scrub down the cork shoes with the scrub brush, removing ingrained dirt. Be sure to re-dip the brush in the liquid, so you're not just smearing old dirt on the shoes.
Wipe down the shoes with your soft cloth again and assess. If they're still very soiled, scrub them down with the scrub brush again.
Allow the shoes a full day to dry. Run a 100-grit sanding block around the cork shoes, sanding them until you've buffed off the surface, slightly dingy layer of cork.