Generating protein sequences conditioned on protein structures is an impactful technique for protein engineering. When synthesizing engineered proteins, they are commonly translated into DNA and expressed in an organism such as yeast. One difficulty in this process is that the expression rates can be low due to suboptimal codon sequences for expressing a protein in a host organism. We propose CodonMPNN, which generates a codon sequence conditioned on a protein backbone structure and an organism label. If naturally occurring DNA sequences are close to codon optimality, CodonMPNN could learn to generate codon sequences with higher expression yields than heuristic codon choices for generated amino acid sequences. Experiments show that CodonMPNN retains the performance of previous inverse folding approaches and recovers wild-type codons more frequently than baselines. Furthermore, CodonMPNN has a higher likelihood of generating high-fitness codon sequences than low-fitness codon sequences for the same protein sequence.