Two excerpts from Nishikawa Joken 西川如見's Chōninbukuro 町人袋 ("Bag [of knowledge] for chōnin"):

Some say that the aversion to using the first, fifth, or ninth month for anything is hard to understand. Even accomplished scholars, when asked, say that they not know the origin of this custom. However, in Buddhist texts the first, fifth, and ninth months are known as something like "months of restraint and simplicity" [斎素月], and in these three months it is forbidden to put criminals to death or kill animals. The details can be found in the encyclopedias [事類全書]. In any case, this is a custom from foreign lands, and though in Japan these three months are particularly avoided, wicked deeds are wicked in any month — while good deeds must surely be even better during such periods. What reason is there for aversion?

As far as I was able to determine, identifying these three months as particularly hazardous for evildoers is a Chinese thing specifically rather than a Buddhist thing generally.

More:

Some say that even among the foolish doings of the children of this age, there are many which were passed down from days of yore for a reason. Here in this Latter Age [of the Law, 末代] some of these original meanings have been lost. In spring, many of the children of chōnin fly kites. The same is true in foreign lands. Even these kites have a purpose: because children are always full of yang energy/heat [陽熱], in the warmth of early spring their ki increases to excess; making kites and flying them overhead causes the children to look up open-mouthed, allowing their internal heat to escape upwards and preventing illness [...]

The more you know!