I've had a lot of luck with the "try to sit your butt on the board" advice combined with the knees bent and arms straight. Another thing I find to work quite a bit is to tell them to count to five before standing up. If they continue to have a problem standing up too early, I tell them not to stand up until they can look at me in the boat and I tell them to do so. Sometimes I even tell them not to stand up at all.
Just yesterday I was teaching a 7-year-old and he was able to get up and ride just fine as long as he didn't stand up. That means he had the hard part (getting up) figured out. Now we just need to work on his balance in the standing position and he'll be tearing things up. I'll probably work with him on shore by using the rope and pulling him up with the board on his feet. That way he can get a feel for the body position he needs.
His sister, who's 4, actually did a little better with the same advice. She got up in the sitting position and was in enough control to look at the boat. We told her to stand and she did so. Of course, she's so short, there isn't a lot of difference between her sitting and standing positions!
She's the youngest rider I've ever gotten up behind the boat.
I'd really steer away from the "push with your legs and pull with your arms" advice because they need to keep their legs bent and arms straight and that seems to tell them to do just the opposite.