Instruction tuning (IT) is widely used to teach pretrained large language models (LLMs) to follow arbitrary instructions, but is under-studied in multilingual settings. In this work, we conduct a systematic study of zero-shot cross-lingual transfer in IT, when an LLM is instruction-tuned on English-only data and then tested on user prompts in other languages. We investigate the influence of model configuration choices and devise a multi-facet evaluation strategy for multilingual instruction following. We find that cross-lingual transfer does happen successfully in IT even if all stages of model training are English-centric, but only if multiliguality is taken into account in hyperparameter tuning and with large enough IT data. English-trained LLMs are capable of generating correct-language, comprehensive and helpful responses in the other languages, but suffer from low factuality and may occasionally have fluency errors.