See my post on "seasonal aspects" of the month of June.
[Don't forget your umbrella!]
The Japanese term for rainy season is bai-u or tsuyu. Bai-u (plum rain) refers to the plums that are just getting ripe around this time. A term used in haiku is "samidare," "the rains of the Fifth Month" (in the old calendar!).
During the rainy season it slowly heats up towards the summer. After each rainy day the temperature edges up a bit. It is moist and sticky everywhere. Some things to be careful of are mildew in your house (open the windows to air it on sunny days) and spoiled food (keep everything refrigerated and carefully clean the area where you prepare your food). Some people get gloomy because of the overcast skies. Many foreign residents fly back to their home countries around this time.
But that is not necessary. On the contrary, when you take some precautions such as carrying an umbrella and extra clothes on longer trips, it is the perfect time for travel. The weather is warm, so you can travel light - a T-shirt, shorts and sandals plus an umbrella are enough. Even on clear days, the sun doesn't have the unpleasant scorching hotness it develops in August. The green of Japan's temple gardens and countryside is at its deepest and most enticing. Sit on a temple veranda and watch the silent garden in the rain. There will be few other visitors to disturb you. The hortensias (ajisai, also called hydrangeas), my favorite flowers, bloom in soft blue nuances. Hakone has mountainsides full of hortensias and is great in the rain (although you won't see Mt Fuji) as are Nikko's deep forests; Kyoto's temple gardens are wonderful and mostly devoid of other visitors, Kamakura, too, is beautiful.
I hear the sound of plums
falling from the trees
dark days in the Rainy Season
ume no otsuru | oto no suru nari | satsuki-yami