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Let $(R,m)$ be a commutative Noetherian local ring. Suppose that $M$ and $N$ are finitely generated modules over $R$ such that $M$ has finite projective dimension and such that $\mathrm{Tor}_i^R(M,N)=0$ for all $i>0$. The main result of this note gives a condition on $M$ which is necessary and sufficient for the tensor product of $M$ and $N$ to be a Cohen-Macaulay module over $R$, provided $N$ is itself a Cohen-Macaulay module.
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