Yeah, OK, literally "It's raining cats and dogs" would be "Дощить котами та собаками" - "It's raining with cats and dogs" (ooh, Instrumentalis, a quite special case)
But yeah, not our expression at all.
Besides "Ллє, як зі відра" you can also say "Пері́щить"! I love this word, періщити, sounds so cool. It literally means to slap, to hit, to smash etc. Cause when the rain is very heavy it's brutally smashing the ground :D