Multimodal machine translation (MMT) systems have been successfully developed in recent years for a few language pairs. However, training such models usually requires tuples of a source language text, target language text, and images. Obtaining these data involves expensive human annotations, making it difficult to develop models for unseen text-only language pairs. In this work, we propose the task of zero-shot cross-modal machine translation aiming to transfer multimodal knowledge from an existing multimodal parallel corpus into a new translation direction. We also introduce a novel MMT model with a visual prediction network to learn visual features grounded on multimodal parallel data and provide pseudo-features for text-only language pairs. With this training paradigm, our MMT model outperforms its text-only counterpart. In our extensive analyses, we show that (i) the selection of visual features is important, and (ii) training on image-aware translations and being grounded on a similar language pair are mandatory. Our code are available at https://github.com/toshohirasawa/zeroshot-crossmodal-mt.