Tennis elbow is an overuse injury that originates on the elbow.  The main cause of this is weak forearm flexors.  In order to mitigate the risk of tennis elbow, you need to implement plenty of forearm extensor strengthening exercises.  However, if you have mild symptoms, you can probably get by (just don't go too hard).  The easiest thing you can do is use a neutral grip for all of your exercises.  This takes significant stress off of your forearms.