Your post made me feel like you live in my house. My oldest is 3 and a half and we have been going through the exact same thing. For about a month I dealt with daily tantrums, bad reports from the nanny and even a phone call from his school. It seemed like he would fight and fight and fight about everything. His tantrums were not just meltdowns where he would scream or roll on the floor but rather I would tell him, no you can't have another snack which would send him running into the fridge to get one himself. He would take action and I would have to physically intervene in order to get him to follow directions. I have also had the same issue of him getting out of time out, and on special bad days sometimes he leaves time out 10-15 times before he just sits and cries and waits for his three minute time out to be over.
I did a couple of things that added value to the situation. First I stopped yelling. I was so frustrated that I ended up screaming at the top of my lungs, like, alot. So I made a concious effort to lower the volume and approach the situation calmly. This actually made him crazier at first, but deflated the situation.
But the absolute best thing I did was to stop asking so many times. I always ended up yelling because I would ask him to clean up maybe 10 times before I got so frustrated that I started screaming. It was a bad cycle. I write a blog, and I posted about my frustrations (see here-- http://thetableofpromise.blogspot.com/2011/02/epic-battle... ) and one amazing commentor said don't ask more than three times. This make a huge difference. A three year old has checked out if you have asked him 10 times to do something. If he doesn't do it by the third time, he is probably not going to do it at all and he is playing the game. That means, move in to either give him a time out, turn off the TV or whatever else sooner. After a couple of times of this early intervention, my son started listening on the first try. But like anything else, he still has his good days and his bad days.
Three is worse that two because you give them so much more credit for rationality. But they still are not rational creatures at 3. And the things that three year olds do are more emotionally manipulative too. I am convinced that this is because they are becoming free thinkers and they don't like all the rules. But they need to learn that following directions gets them the freedom to do the other things that they want. Likely this is not a situation that will completely go away. My son is smart and energetic and independent. Likely his independent thinking will manifest in other ways as he gets older. I have abandoned the notion that this is a phase. I am just trying to use this as an opportunity to teach him that he will ultimately get what he wants if he listens and plays by the rules.
So far so good....Good Luck and hang in there!!