Kokuu translates as "Grain Rain", and is the sixth of the Nijuushi Sekki. Summer is coming! The rains will help the harvests grow.
Prepare for a grey sky, and carry an umbrella. Or find plenty of things to do inside if it isn't sunny out. If I have nothing to do, I tend to go through things I haven't looked at in awhile and sort through what to keep and what to let go, or organize finer points like the DVD rack, since I don't normally take the time to do it. Bring some flowers inside, light some candles, have a cup of tea. Read a book. Whatever you want to do!
If you really like, there are plenty of books at your local library. Just use the catalog to look up keyword 'japan' and see what's there! I found several old photo compilations of Meiji-era geisha, books on Japanese sewing crafts (written in English), and cookbooks. In the childrens' section, fairy tale stories and kids' books like "Sadako + the Thousand Paper Cranes". Never discount childrens' literature if you don't have kids- some stories do not need to be in-depth and complex for them to be enjoyed. If you prefer something more thrilling and adult, perhaps Laura Joh Rowland is more your speed.
If it's wet out, I would avoid silk and stick to wool, cotton, or hemp. Of course, synthetics are what most people own today, and those will work just fine. Down here in the South, we don't really have 'seasons', per se, other than 'hurricane season'. For me, rain means subtle colours like dark green, greys, blues, and tans. I would convert to wearing hitoe kimono which are not transparent around now, although technically, it's still awase (lined kimono) season. The reason is that it is unbearably hot outside. If you're just hanging out around the house, yukata or jinbei are immensely comfortable! <3