This is a good book on understanding 3 year olds.
"Your Three-Year-Old: Friend or Enemy" (Paperback)
by Louise Bates Ames (Author), Frances L. Ilg (Author), and you can get it at Amazon.com
3 years old is a hard age, and developmentally they are changing a great deal. It's on the cusp of "being a big boy" but yet, they are still "baby" too... and often they get stressed & frustrated with the littlest things.
Still, tantrums are not easy. They are having to learn so much and are expected to follow rules and "behave" 24/7, which they can't do all the time.
In any event... for us, what we did is always talk about things with our daughter... and just letting her "vent" when she needs to, while at the same time explaining what "feelings" are... and how even grown-ups have "tantrums" too... BUT, we are all a "family" and we try our best not to take it out on each other etc. We are are a "team..."
Then, we also tell our girl, that if she is feeling grumpy or anything, she can go in her room and scream/yell all she wants... and it's okay. THEN, when she feels better, she can come out... and it's fine. It works for us. Sometimes she tells us she doesn't know WHY she feels grumpy, and sometimes she does have a reason... but at least she "knows" she has a safe place to go and just let it all hang out, so to speak. So it works for her, and us.
The thing is, children cannot always censor themselves or modulate their emotions, nor monitor themselves perfectly...but if we navigate them and "teach" them about HOW to respond to frustrations/stress, then it teaches them how to manage it... and as they get older, they get better at it. It also teaches them "problem solving".... ie: I am feeling grumpy... um, what can I do? Can I talk about it to Mommy? Can I go in my room and let it out... can I yell at somebody? Can I just hit my pillow instead? etc. YOU teach them different ways of letting it all out. ANYONE, child or adults, needs to learn how to "vent", right? Heck, even adults don't know how to do that pleasantly sometimes, right? Well same for children. This is how we teach our kids.
And yes, they also need to learn that screaming is not the way to get things... and that yes, sometimes Mommy will not just give me what I want. That's okay too. Pick your battles.... or you will go nuts. When my girl/son screams...sometimes I simply tell them "Mommy's ears don't work when you yell or scream...." then I just keep doing whatever it is I'm doing. And, there comes a point when they just deflate themselves... and then the next thing you know, they are in a regular mood again.
But through it all, they need to know HOW to manage even their tantrums... and I believe, instead of just making it a "behave or else" kind of thing... that children benefit by being taught how to handle their feelings/emotions too.... and that it's okay to have these feelings. In the long run... a child that is "comfortable" with understanding feelings/frustrations/emotions, will have a more articulate response to it, AND in figuring out other people. We have seen this in our own daughter... she is very articulate and "wise" about understanding people/their facial expressions/their feelings and what is right and wrong. And, she is confident about handling her own frustrations now.
It takes time, and consistent communication and "understanding" of them.... and teaching them more positive ways of grappling with frustrations. BUT, it's a process. And in the immediate outburst... sometimes you just have to use several methods at the same time... ie: "use your inside voice" while explaining that if he continues to scream, you will not play with him. Or, "that's not the way to ask for something... how can you do it nicely..." or, "Come, let's go in your room, and when you feel better, you tell Mommy all about it...." Or, "When you stop screaming... Mommy will be right here, and then we can have a snack..."
Also, what really helped our girl too, is by explaining that we don't expect her to be "PERFECT." That it's okay to make mistakes, but the main thing is she try her "best" and we always love her no matter what. That if she tries her best... and that's the best she can do, that's great! We "see" that she IS doing her best, and even if it's not in the manner that "we" want... it's okay, and we have to realize that for the child... that this IS their "best" and they have tried. End of story. No need to keep nagging them about it either. They have come full circle.
Well just some ideas and what I do with my kids. Hope it helps, sorry for rambling on and on...
Best of luck,
Susan