Spring and Autumn are considered by most to be the optimal pilgrimage season. The Spring season is between mid-March and the end of May. Most pilgrims choose to schedule their pilgrimage from early April to May. The weather from early- to mid-March can still be a bit too cold, with temperatures being only a few degrees above zero on high grounds. On the other hand, if you wait too long, Japan's rain season begins in June and lasts for several weeks. However, if you are willing to brave a little chilliness to watch the cherry blossoms (sakura) unfold into a sea of pink while you walk the trail, the sakura season in Shikoku lasts from mid to the end of March. Click here for the forecast of the next cherry blossom season.
If you're still not persuaded, Japanese stores and resaurants often offer sakura flavored yum yums during sakura season.
Summer is not the ideal season for a pilgrimage. Japan's rainy season (tsuyu) lasts for several weeks, beginning in June and continuing until mid-July. Execept for the northern most island of Hokkaido, almost the entire Japanese archipelago is affected. The rainy season features many sudden downpours that often feel like free-falling buckets of water, making it perhaps the most unpleasant time to visit Japan during the year. On the bright side, visitors are scarce, and places such as temples, gardens, and hot springs can be quite the sight on rainy days. For the brave souls who choose to go anyway during this time: be sure to bring rain gear that is fully rain proof, because even just being in heavy rain for half an hour can completely soak the contents of your unprotected backpack. In addition, the intense humidity can find its way into your things even if the rain doesn't. I found it very helpful to seal my things into air-tight large ziplock bags. More details on this can be found on the What to Bring page.
The warm and humid summer months provide the perfect environment for mosquitoes to grow into adulthood. They continue to hunt for blood until late September/early October, and are the most active at dusk. A breed of large, striped mosquito called the tiger mosquito can spread a disease known as Japanese encephalitis, which causes fever, headaches, and vomiting for up to two weeks, and is fatal in rare cases. In addition to bringing rain protection gear, you should try to also be as mosquito-proof as possible. Even with full protection, it is inevitable that a few rebels will nonetheless get to you, so I would advise also bringing products for mosquito bites. Check out What to Bring for a list of recommended products.
Due to reasons stated above, pilgrims are not advised to plan their pilgrimage for the hottest months during summer. While the rainy season is less of a problem if you are well-equipped with waterproof everything, the 4-6 weeks after that is simply unbearable heat, with a mosquito problem on top of it all. Compared to most places in the United States, summers in East Asia come with an incredible amount of humidity. It would certainly feel like hell on earth to hike and trek for hours a day in that weather. Specifically, the months of July and August should be avoided.
Autumn in Japan is between mid-September and mid-November. The scenary and weather during this time is simply delightful, making autumn the second most popular season for pilgrims. The ideal time to begin would be the beginning of September, and you should plan to finish before mid-November when temperatures at higher elevations may occasionally fall below freezing point.
Winters in Shikoku are rather mild compared to the rest of Japan. You will rarely see snow except when you hike up mountains. Nonethless, most advise against going during winter. The days are short so you may find yourself often walking in the dark. Moreover, because winters are unpopular for pilgrims, businesses may choose to close during this time, or temples may close of sections of the temple grounds for reconstruction for maintenance purposes. Nonetheless, I've read about several stories of pilgrims who completed the full pilgrimage during winter, or even prefer that time of the year due to the tranquility and beauty of it.