Two approaches:
Search tourist websites for the countries you are interested in. They very often have average data on temperature and rainfall by month.
Just Google countryname monthly rainfall. You may get general descriptions instead of hard (tabular) data, but since you only have to collect the data once, that's doable.
But as you get from the comments below your post, how useful that data is remains doubtful.
As an example: The first hit for the austria monthly rainfall search says:
... altitude determines the precipitation pattern; while high-level areas in the Alps may have a high average rainfall in excess of 2000 mm per year, while some regions in the flatlands of Austria have only 600 mm annually.
(emphasis mine)
Note that this page mentions the WeatherOnline (UK) Climate of the world page as its source. There you have descriptions for many countries.