I have a 3-year old boy and 1.5-year old girl, and a (overall friendly) cat. As infants both kids were rough to the cat, with hair pulling and hitting, although each was different in what they did. Once they could walk, then chasing the cat was a fun activity. They have very little self control and get excited when the animal is around.
Each kid is different; the girl at 1.5 is a lot gentler with the cat than the boy was at the same age. In fact, she's about as gentle as he is at 3! I chalk that up to different natures.
Throughout this, we've done all the various things others have suggested, but I think the thing that helped most was just growing up and gaining maturity. Now that the boy is 3, he takes pleasure in seeing her sit on his lap while he pets her.
There have been a few cases where the cat didn't like something, and while we haven't had any scratches or bites (it's a very self-controlled cat), there were a few close calls. Those gave us opportunities to teach the boy about the cats claws and teeth and how she protects herself. He got (accidentally) poked by a claw once, and that seemed to wise him up quickly as to the consequences of scaring the cat.
At 3, the question of the day is always 'why?', so use it to your advantage. Why do we have pets? Why does the pet live here? Why does it chase mice (or cats)? etc. etc. Sit with the child and observe the animal quietly, and explain what it's doing, and why. And +1 to the other answer to role model how to handle the animal properly, and why.