Multitran gives the following examples:

- быть недовольным кем-л.
- сердиться на кого-л.
- отчитывать
- пилить

Of those four, the last two seem the most fitting. "Быть недовольным кем-л." means to be dissatisfied with someone, and "сердиться на кого-л." means to be angry at someone. Neither of those two have the conversational style of "getting on someone's case". The last two are more conversational and, according to the dictionary entries I've read, denote the action of criticizing/reprimanding another person for an extended period of time.

I'm sure some native speakers have other examples, though!