ADHD is extremely hard to diagnose in a 3 year old. I'd go as far as to say it's flat out irresponsible for a doctor to even attempt to make a diagnosis at that age, no matter how smart your child is. It's even more irresponsible for that doctor to have put your 3 year old on meds!
Before an accurate diagnosis of ADHD can be made, a child has to be both mentally and physically mature enough to NOT display ADHD symptoms. I don't know any three year old who has that maturity (being smart and being mentally mature are two completely different things).
Also, there are several questions you, as a parent must ask yourself (and answer honestly).....
-Have you explored dietary changes? What does he eat every day?
-Are you making sure he gets enough sleep for his age?
-Are you making sure he gets a good 30 minutes of exercise time at least 3 days a week (where he can run around, ride a bike, climb, kick a ball, etc....)?
-Do you ever tell him no?
-How do you react when he throws a tantrum?
-Does your reaction encourage him to throw more tantrums?
-If you ever do tell him no, then when he nags or throws a fit, do you eventually give in, and let him have whatever you originally said no to in order to get him to calm down?
-If you ever do tell him no, does he do it anyway, and if so, do you just ignore him because you don't want to set him off?
-Do you ever discipline him? If so, how?
I have to ask these questions, because I have a sweet, dear cousin who is raising a monster. She had the monster diagnosed with ADHD, but the truth is, nobody in the monster's house has ever told the kid no. I'm serious. They are afraid of him and what he will do if they say no to him, so he has no boundaries. Sad thing is, he is afraid of everything, and will have a complete freak out, raging anger fit if he can't have what he wants. He is violent, throws things, breaks things, hurts his younger siblings, hurts himself.... it's horrible. But the poor kid has never been taught any differently. Unfortunately, my sweet cousin has no idea that she is doing this to him. She and her husband read a lot of books from, "experts," and the, "experts," say that discipline hurts a child's creativity, self esteem is based on parents who indulge their children with talk of feelings and letting their children have opinions, and choosing battles, and that punishing a child is what bad parents who are on power trips do, and other such ridiculous nonsense.... in my opinion, the experts are morons! I have a lot of very firm beliefs on those subjects, proven methods, and tons of great ideas based on personal experience (not experts!), but now isn't the time.
Back to your son.... I'd definitely check his meds. If his tantrums have gone from bad to worse, it's probably the meds doing it. I'd get him off the meds, away from doctors, and start over at home. Start with what you are feeding him. Go natural. Not necessarily organic (unless you can afford it), but natural. Whole grains breads, whole grain pasta, whole grain cereals.... sugar isn't horrible, but keep it in the lower numbers and eliminate anything with high fructose corn syrup (did you know most ketchups have HFCS?), and anything with artificial dyes (RED IS THE WORST for ADHD kids). More fruits and veggies (especially green stuff), and white meats over dark. Nothing fatty or fried, nothing with ingredients that you can't pronounce, nothing with artificial ingredients. No fast food, no candy, no soda, natural fruit juice, and tons of water. Follow this for a few weeks, maybe a couple of months, and see what kind of difference it makes. Keep a journal of what you are feeding him, and what kind of day he has. Read ingredients on EVERYTHING. Medicines and even vitamins have artificial dyes in them, so be careful. After a couple of months following this regime, you can slowly introduce other foods back into his diet, but each time you do, write it down and watch how he reacts. You will start to see a pattern in how what he eats affects his moods and self control.
Make sure your son is getting enough sleep. At 3, he should still be getting between 12-14 hours in a 24 hour period (11-12 at night, and a 1-2 hour nap during the day). If he seems wired at bedtime, you're most likely either feeding him something for dinner that he reacts to, or you are missing the window where he should be going to bed, and letting him slip into that period where he's tired, but his adrenaline has hit and he can go another couple of hours, at least! It's hard to find that window, but for a three year old, it's generally somewhere between 7-7:15pm.
I could go on and on, but I'll stop here and listen to your answers first. It makes it much easier to offer more specific advice when those few questions I listed above are answered. I hope some of this could help you.
I have had a good 20-25 years working with difficult children. 10-11 years were spent teaching in daycares where they'd bring me the tantrum prone children, because I do know how to handle them. Those kids ended up being some of my most favorite kids, and I still keep in touch with many of them.
I have 4 children of my own, and 2 of them, I suspect have ADD. My oldest is 8, and I'm 99% sure she does, but I won't get her diagnosed unless I have to.... we're treating her right now with diet changes, exercise and good sleep.... so far it's working! She used to take a supplement, but she hasn't had to since before Christmas, and she's doing great in school. I have a 3 year old who could potentially have ADD, but she could also be just acting like a 3 year old. They both react horribly to red dyes, and high fructose corn syrup, so that's something I really have to stay on top of! You'll be shocked at just how many foods have red dyes and HFCS in them! Even yogurt does! I always thought yogurt was great for them, until I started learning! Also, I have ADD. I wasn't diagnosed until I was in my early 20's. I have learned a lot about the subject over the last 15 years..... I know a ton of ideas that work to help ease the symptoms of ADD, and I know a ton more that don't work at all!
Sorry to ramble on and on.... I do tend to love the soap box!