In reply to iksander:
Given it's both wrists the probability lies in biomechanics-overload or systemic problem, most likely biomechanics. It's possible you do too many handstands and push ups which pressure sensitises your wrists to problems when you crank open handed. It may also be sensitised by something else, occupation etc. Open hand climbing should not hurt. In this case easing off on some of these activities may help. Usually it's a weakness of the wrist extensors, which then get dominated by the flexors causing an imbalance and problemos. Sometimes it's weakness both extensors and flexors. Theraband or dumbbell wrist curls is the likely answer. You want stability, and strength endurance so start 3-4 x 10 but increase by 5 reps to 3-4 x 20-30. When you are better you can work strength and power. Daily initially then decrease frequency as you feel you need more recovery. Maintenance 1-2/week. That failing, doctor for x-ray and full blood tests to check for systemic problem.