I intellectually understand 100% how to juggle three balls continuously. Despite this theoretical knowledge, I am still utterly unable to juggle three balls continuously. Alas, such is life.
The pattern used by the hands to juggle three balls is actually very simple and rythmic. Basically all I knew before going into this is that you always have one hand throwing, the other hand catching. Each hand alternates without stopping: throw, catch, throw, catch. The way all three balls stay in the air and the hands are free to catch and throw is by maintaining this pattern; if the juggler stops, he must hold two balls in one hand.
The way to achieve a system where both hands can easily catch, throw, catch, throw is to have each arm going in little circles. At the bottom of the circle, towards the middle of the body, your hand comes upward to throw the ball. At the top of the circle, towards the outside of the body, your hand is coming down and catches the ball to carry it around the circle, building up momentum for the next throw.
Besides drawing the circles for the hands to follow, color coding the balls helped immensely in ensuring the pattern was not lost. In my first try (not shown above) I drew three balls but quickly lost track of which was which, and what direction they were going. In the second try I incorporated the circles, which helped me remember which direction each ball was going (if you pay attention, each ball follows a little infinity sign pattern from hand to hand), but I could not remember which was which and where the animation was supposed to loop. In the third and final attempt (for now at least!) I was able to follow each individual ball all the way around the looping animation.