While many do it, it is inadvisable for the deer hunter to take running shots. Running shots often result in an injured deer escaping to suffer either a long painful recovery or a slow death and should be reserved for follow-up attempts to stop an injured animal from escaping.
However for the photographer running deer offer an opportunity to capture a different kind of image with no harm to the subject. To successfully shoot running deer one must pan with the subject. For shooting long telephoto lenses a good gimbal head is virtually a necessity. The images in this post were made with the Canon 600mm lens mounted on a Kirk King Cobra head using panning and image stabilization.
While it is imperative to pan with the running deer, panning is no guarantee of success for not only does a deer move horizontally while running but it moves vertically as well. If you try panning with deer you will find that most images will be ruined by the vertical movement however with a little luck an image or two just may make the grade.
As light levels increase so can shutter speeds. In this image you see that the buck is nearly frozen by the combination of panning and the 1/160th shutter speed.
So to recap, if you encounter a running deer and your weapon is a rifle, hold your fire: the ratio of risk to reward it just too great. However if your weapon is a Canon (or a Nikon for my friends on the dark side) pan and shoot rapid fire. You may make your best shot of the season!