Is your saddle level or pointing downwards at the nose? Some people put their saddles up too high, which creates pain in delicate areas. They then point the nose downward instead of lowering the whole thing, which means they are sliding forwards onto the bars and putting more weight than they need to on the arms and wrists.
Riding with stiff straight arms is another culprit. You want a little bend in your elbow to soak up road shock. If your top tube and/or stem are too long, that can force a straight, rigid arm.
Change positions frequently when riding. Drop bars were designed with this in mind. If you don't want drop bars, lift one hand off at a time and stretch out from time to time.