The Macmillan English Dictionary for Advanced Learners rightly describes the idiom it’s raining cats and dogs as old-fashioned, but there are plenty of other idioms associated with different types of weather that are in current use.
The expression under the sun, for example, is used for emphasizing that something includes a very wide range of things, e.g. 'She can talk on any subject under the sun’.
If you are snowed under, you have so much work that you are unable to deal with all of it.
If you get wind of something, you find out about something secret or private, as in ‘The authorities got wind of a plan to smuggle weapons into the country'.
If you have a face like thunder, you look extremely angry, while something that moves like greased lightning moves very quickly indeed.
If you find yourself under a cloud, your reputation has been damaged because you may have done something wrong.
If you are as right as rain, you are healthy or happy again after an illness or an unhappy experience, e.g. ‘She looked as right as rain when I saw her’.
Something that goes down a storm is extremely popular, as in 'The play went down a storm on Broadway'.
If you put a plan or an idea on ice, you delay doing anything about it.
Finally, the word weather itself gives us the expressions under the weather (not feeling well) and in all weathers (even when it is raining, snowing etc), as in ‘He goes out jogging in all weathers’.