  Fuglede's spectral set conjecture states that a subset $\Omega$ of a locally
compact abelian group $G$ tiles the group by translation if and only if there
exists a subset of continuous group characters which is an orthogonal basis of
$L^2(\Omega)$. We prove that Fuglede's conjecture holds on cyclic groups
$\mathbb{Z}_{pqr}$ with $p,q,r$ distinct primes.
