Always using the same grip in a strength exercise will not allow you to fully develop the target muscle. Best is to use a variety of grips to produce fuller and more varied development. For example, a pronated grip press-down targets mainly the medial head of the triceps. A supinated grip targets mainly the lateral head while the neutral grip targets mainly the long head of the triceps. Doing a pull-up with a pronated grip targets mainly the upper lats and a supinated or neutral grip targets mainly the biceps and the lower lats.
The exact grip to use is important when working with athletes, especially higher level athletes for specialized work. In general physical preparation, it is not that important, but when you get into the phase of specialized training and use specialized exercises that duplicate what occurs in the skill execution, the type of grip becomes extremely important.
For example, in baseball batting, for a right-handed batter, the right hand is supinated while the left hand is pronated when gripping the bat in the swing. During the wrist break, the grip is neutral. Thus, this is the grip that must be used when duplicating the triceps involvement in hitting. In the follow-through once again there is supination-pronation.
Tennis players must isolate the grip when strengthening the triceps to duplicate the different types of serves. For example, a slice serve requires more of a neutral grip and even the flat serve requires a neutral grip in the early phases and turns into the pronated grip immediately before making contact. The same occurs in the volleyball spike and other throwing and hitting skills.
Suffice it to say that you should do an analysis of the game skills to determine the exact joint action involved. Then you can create exercises to duplicate the same joint action and develop strength as it is displayed in execution of the sports skill.
See Kinesiology of Exercise for more information on exercise execution and the different grips that can be used. For sports-specific exercise information, see Explosive Running, Explosive Basketball Training, Explosive Tennis, Women’s Soccer: Using Science to Improve Speed, Explosive Golf and Build a Better Athlete. by Dr. Michael Yessis.