Good points above about relaxing the grip and straightening the wrists.
--Relax your face and neck. Let them soften and feel like they droop (but they don't, really.)
--Relax your shoulders and let them drop down & back.
--Ride with your spine straight and your trunk relaxed, knees just lightly against the tank, balls of the feet on the pegs and legs relaxed.
--Relax your arms and allow your elbows to drop down.
--Relax your fingers, hands, wrists. How loose of a grip can you hold on the grips and retain control (and control will be enhanced with a relaxed grip).
Next, re-relax. It is natural to tense up and not realize it. Consciously relax.
Try this for-free effort to relieve your wrists. If that doesn't work, then consider spending money for different bars or a medical evaluation of your wrists.
This^^^,+++ a quick stretch every time you stop, concentrate on your thumb, the downward motion of your wrist, and the upward motion of your wrist, 20 to 30 seconds on each hand/area at a time...while grasping your fingers at the top of the other hand slightly pull them backwards this will stretch your palm and wrist area too, repeat in reverse.
With your left arm at your side, tilt your head to the right side using the opposite hand placed over the top of your head slightly pulling toward the right side, extend your fingers on the hand at your side and slowly move your hand toward your back to increase the stretch, do each side (reversed hands/arms) 20/30 seconds, you might want to repeat a couple of times...try to remain relaxed and have your body in a straight stance, do not let your head droop, stand tall, or the stretch will not be as effective....Again, relax while stretching, this is key for your tendons and nerves to work properly, and for the blood to flow to the areas needed.
I hope this makes sense, and helps...