Flat bars are causing that. You're keeping your hands in an unnatural pronated position for too long, and the road shock only exacerbates the sensation. Try out some bar-ends and see if that helps. Having 2-3 different hand positions that you can change up from time to time can really make all the difference in riding comfort.
Also, learn to grip the handlebars with less force. If you use a death grip, ease up and use a firm, but gentle grip instead.