What's the best way to relieve wrist pain from typing too much?

Breaks and stretch. Yes, frequent breaks...Even if they are just for a minute or two can help. Ergonomic correct positioning is essential. In addition warm up flexibility exercises and stretching during quick breaks can make a big difference.
Breaks. Frequent breaks and occasional icing can help with typing related wrist pain. An ergonomic workstation, keyboard and mouse are also essential to prevent other injuries as well.
Have you considered. Voice-activated dictation software? ("naturally speaking", and others.) let your wrists rest!