Even if I visited a few other very hot destinations, such as Las Vegas, Hong Kong or Puerto Rico, I didn't really know what to expect before traveling to Japan in summer for the first time.
Actually, I'll have to be clear and drop this: Japan is very hot in summer! During my stay in July and August, it sometimes went up to 36°C in the shade, and of course temperatures felt even stronger under the sun. At night, the heat rarely falls below 27°C. To sleep, air conditioning is quite essential and fortunately, it's provided in almost every accomodation.
During the day, shops, trains and buildings also provide cooling through their A/C ("aircon" in Japanese). Every summer, salary-men are entitled to a "summer wear" and can, if desired, drop the tie and exchange their shirt for a blouse.
Japanese people are traditionally discreet and don't often show they suffer from the weather. Their body is certainly more accustomed to undergo heat waves in summer, but they also prepare accordingly. You'll see a lot of fans, sponge-handkerchiefs and umbrellas to hide from the sun. Some women also wear long gloves.
If you are traveling to Japan during summer, be prepared to sweat a lot. Uniqlo has a "dry" range regarding this con. But it's not unbearable either. Of course, it depends on your heat resistance, but I don't see how that could prevent you from visiting Japan. I was almost more bothered by mosquitoes, especially in parks and japanese gardens. Note that daylight hours are significantly different from our western zones. In Tokyo, the sun rises at 4:30AM and sets at 7PM.
Visiting Japan in summer also means masturi festivals and their fireworks, ice creams, cicadas and beaches such as Enoshima, Amanohashidate, Fukuoka... In almost every city, it's possible to get some fresh air: just outside Kyoto for example, Arashiyama, Kurama or Fushimi Inari will amaze as they refresh you!