I own a D300. Have owned it for over a year but have never used Dynamic Area AF. Always used either Auto Area or Single. When I tried Dynamic Area AF the chosen focus point stayed where it was. Even when the subject (my dog) moved from under the focus point. Should one of the other 51 points automatically become active? As long as my dog is in the frame? What should I be seeing happening in the viewfinder?
When I switch on Focus Mode C then the selected focus point tracks the subject. As "predictive AF" is automatically used.
To properly take pictures of a moving subject with your D300, you have to set your camera to Continuous Focus Mode (AF-C) like warprints stated and in that mode you will not see focus point displayed in your camera. I shoot my BMX and other action in Continuous Focus Mode (AF-C) only with 9 point dynamic(some times single point), for my shooting more points makes the area that the camera will look at to wide. But try different settings (custom a3 adjustment) 9, 21 , 51 or 51 3D tracking(slowest) depends how far to the sides that moving objects needs to be in focus.
BTW your camera after you master these settings is simply fantastic at maintaining AF