Tokyo is massive. I'd argue that three days isn't enough.
You want to go to places like Shinjuku, Shibuya, Harajuku and Ueno to see the standard Tokyo sights and sounds. Like London, each area is different and offers a different experience. It's a busy place, but other than the size and the number of people it's easy to get round. All the signs at the stations are in English as well as Japanese, even out in the sticks. The trains are amazing there, and even though you have heard about how on time they are and how well run it is, it still surprises you!
If you're into your electronics, a trip to Akihabara is a must. Saying that, it could be a lot different now as it was fifteen years since I was there. But it used to be the place to go for the latest consumer electronics stuff and it was an amazing place, especially in the evening when it's all lit up.
When are you going? The best time is Spring or Autumn as the weather is fantastic then and the scenery looks the best. Autumn probably my favourite time there.
Kyoto is a great place to go for the temples and shrines, some of the gardens are amazing. Nara is also good, but you probably don't need to do both.
Osaka is also worth seeing, and more interesting as Japan's second city than Birmingham is here
Hiroshima is definitely worth a visit. The Peace Memorial Park and museum is an experience I'd recommend. There's lots more to Hiroshima as well, and again wherever you go there's amazing scenery to enjoy.
Depends exactly what you're after. I was out there for a few years and didn't see all the places I wanted to see, the same as in the UK I guess. I'd say make time to go to an onsen (hot spring baths), ideally one in a mountain area, possibly near Fuji. Even better if you go when there's snow on the ground.
Try and stay in some traditional ryokan as well as western style hotels. I know you like your food, and the food there is amazing. Obviously sushi and sashimi should be on the menu, but there's a massive range of food available.
Never went to Okinawa, but it looks amazing. Quite a journey out to Naha though if you're only there for twelve nights. It's like a different country I've been told.
It's also worth looking at visiting some of the other main islands of Japan. Honshu is the main one where Tokyo, Osaka, Kyoto and Hiroshima are. But Hokkaido in the north is breathtaking in terms of the scenery, as are Kyushu and Shikoku. But as your time's limited you might want to stuck to Honshu.