Recent work has questioned the importance of the Transformer’s multi-headed attention for achieving high translation quality. We push further in this direction by developing a “hard-coded” attention variant without any learned parameters. Surprisingly, replacing all learned self-attention heads in the encoder and decoder with fixed, input-agnostic Gaussian distributions minimally impacts BLEU scores across four different language pairs. However, additionally, hard-coding cross attention (which connects the decoder to the encoder) significantly lowers BLEU, suggesting that it is more important than self-attention. Much of this BLEU drop can be recovered by adding just a single learned cross attention head to an otherwise hard-coded Transformer. Taken as a whole, our results offer insight into which components of the Transformer are actually important, which we hope will guide future work into the development of simpler and more efficient attention-based models.