I am seeing a lot of posts about 3yo's on this board! so you are not alone, mama. This is a rough age. My 3yo DS has been a handful too to put it mildly.
When my DS has a hard time controlling himself I have taken to either time out in a TO chair or "calm down time" in his room. For me, hitting is a serious offence. That warrants a TO. He knows if "you hit you sit." I only use TO for a few designated behaviors. Hurting people, swearing at me and purposeful destruction of property are my big things. (ie when he pushed over the TV because I told him he had watched enough
I know not everyone does TO but honestly I have no idea how you don't have any punishments without completely loosing it yourself. I know that I need to remain in control or I will want to hit and throw a tantrum myself.
If he gets really out of hand but is not hurting anyone, tantruming and screaming at me for example, I generally tell him he needs to go into his room to calm down. If I remember, I ask if he would like a hug first. Sometimes a hug is enough to console him, but sometimes not. If he goes to his room, I tell him he can read a book or play or do whatever he wants to calm down. I tell him he may come out of his room when he is ready to be respectful. The first few times I had to get him to go back in his room a couple times before he really calmed down, but he is pretty used to it now. I feel like I am trying to teach him to walk away before he does something bad. He has very BIG emotions and I want him to learn how to manage them. I do not want a teenager punching holes in my wall because he cannot control his reactions to his emotions. (DH was like that when he was younger.)
Oh and we also do a lot of verbalization... like "I understand that it makes you angry when mommy says you cannot have a cookie. You really want a cookie! It is ok to be angry and tell mommy you are angry. But it is never ok to hit mommy."
Not sure if any of these suggestions are helpful, but it's the best I've come up with. I think it's really important to remain in control and not let his emotions scare you. It must be scary for them to feel so out of control and they are looking to us to set the boundries and love them through it. GL!