▲on rainy days和in rainy days都可以说。例如：

She usually goes to work by bike except on rainy days. 除了雨天之外,她总是骑自行车上班.

《现代汉英综合大词典》

It’s better to do some reading on rainy days.  在雨天读书会更好。

Umbrellas and raincoats are what people need in rainy days. 伞和雨衣都是下雨天人们要用的。《简明英汉词典》

She often feels blue in rainy days. 在雨天她经常忧郁寡欢。

▲我认为，rainy days既可以看作“具体的一些天”，也可看作一个“时间段”，所以用on和in 均可。