There is a slightly less impressive but much easier way to juggle three in one hand & that is by using a Column Multiplex throw, which reduces the pattern to a Two in one hand juggle.
Start by holding two balls in the hand you want to use for the pattern & one in the other. Throw the single ball as a high cascade throw over to the hand with two balls. As the first ball peaks, throw the two balls straight up as a column multiplex underneath it. The topmost ball of the multiplex should peak at the same height as the cascade throw. Catch the first ball with the same hand & toss it back to your other hand as another high cascade throw underneath the multiplex. Then catch the lowest ball of the multiplex, then the second - both in the same hand that threw them, one after the other. Then finish by catching the single ball in the opposite hand & stop.
Keep practising this bit until you feel comfortable with it. The three throws of the exercise can be thought of as the start of the pattern (the first throw), one throw of the two in one hand juggle (the multiplex) & a finish (the last throw).
Once you can control those three throws start adding a few more in the middle so that your chosen hand juggles all three balls for a few throws. If you have read my Two in one hand tutorial, you'll remember I talked about three different patterns that you could use, inside, outside & columns. You can juggle three with a multiplex using any of those patterns, but the columns pattern is the easiest to learn with in the beginning. Inside & outside requires you to throw the balls around each other, which is okay when just throwing one ball at a time. When throwing two at the same time though, it is easy to throw the balls so that one goes much further in one direction than the other. This makes things difficult when you need to catch them. It is easier to control a column multiplex if it is a straight up & down throw.
Start again with two in one hand & one in the other & start the pattern by throwing in the single ball. Throw the multiplex straight up on the inside as before then catch the single ball, BUT DON'T SCOOP YOUR HAND UNDER THE MULTIPLEX!! Instead keep your hand on the outside & throw the single ball straight up to peak at the same height as the topmost ball of the stack. Catch the lower ball in the base of your hand then the top ball with your finger tips. Wait until the single ball peaks then throw another column multiplex straight up on the inside. Catch the single ball & make a cascade throw underneath the column, catch all the balls & finish as before.
Keep practising, throw one ball in, juggle two/three in one hand for a bit then throw the single ball out again. The more you practise the longer you will be able to run the pattern.
Remember to scoop your arm, some people tend to ignore the scooping action needed when they start juggling with multiplex throws. Without scooping you have to put more physical effort into each throw, which affects the accuracy of the throw & your arm will soon get very tired. Also when throwing the multiplex keep your attention on the topmost ball. Go back to the Column Multiplex tutorial for the explanation why!
Once you have mastered juggling three in one hand with a multiplex throw columns style, try the inside & outside patterns as explained in the Two in one hand tutorial. Also make sure you learn the trick using both hands.