Yuichiro Fujiwara's comments seem to answer the question in full so I am making it an answer. I quote below from <a href="http://cms.math.ca/cjm/v29/cjm1977v29.0255-0269.pdf">Kronecker products and local joins of graphs</a> by M. Farzan and D. A. Waller, <i>Can. J. Math.</i> <b>29</b> (1977), 255–269. >By a <i>1-contraction</i> of $G$ we mean the removal from $G$ of each vertex of degree 1 (and its incident edge). > > 5.3 THEOREM. Let $G_1$ and $G_2$ be connected graphs with more than four vertices. Then $G_1\times G_2$ is planar if and only if either > > (i) one of the graphs is a path and the other one is 1-contractible to a path or a circuit, or > > (ii) one of them is a circuit and the other is 1-contractible to a path. This leaves open only the case in which at least one of the graphs has fewer than five vertices. Proposition 5.4, which I won't bother reproducing here, covers some of these cases. <a href="https://hal.archives-ouvertes.fr/hal-00387303/document">Beaudou et al.</a> addresses the case $G=K_2$. A complete solution appears to be still an open problem.