Since your hand and wrist are connected to your arm, you need to move your arm up and down to move your hand from string to string, but not always. Sometimes, depending on what it is you are playing, you can only move your wrist to change strings.
If you are playing "Don't Fear The Reaper" by BOC (great tune) you should be able to leave your picking hand in one spot and only move the wrist up and down.
Hope this helps
-U