While conversational semantic role labeling (CSRL) has shown its usefulness on Chinese conversational tasks, it is still under-explored in non-Chinese languages due to the lack of multilingual CSRL annotations for the parser training. To avoid expensive data collection and error-propagation of translation-based methods, we present a simple but effective approach to perform zero-shot cross-lingual CSRL.Our model implicitly learns language-agnostic, conversational structure-aware and semantically rich representations with the hierarchical encoders and elaborately designed pre-training objectives. Experimental results show that our model outperforms all baselines by large margins on two newly collected English CSRL test sets. More importantly, we confirm the usefulness of CSRL to non-Chinese conversational tasks such as the question-in-context rewriting task in English and the multi-turn dialogue response generation tasks in English, German and Japanese by incorporating the CSRL information into the downstream conversation-based models. We believe this finding is significant and will facilitate the research of non-Chinese dialogue tasks which suffer the problems of ellipsis and anaphora.