  In \cite{BSV}, Borisov, Salamon and Viaclovsky constructed non-standard
orthogonal complex structures on flat tori $T^{2n}_{\mathbb R}$ for any $n\geq
3$. We will call these examples BSV-tori. In this note, we show that on a flat
$6$-torus, all the orthogonal complex structures are either the complex tori or
the BSV-tori. This solves the classification problem for compact Hermitian
manifolds with flat Riemannian connection in the case of complex dimension
three.
