  A threefold isogenous to a product of curves $X$ is a quotient of a product
of three compact Riemann surfaces of genus at least two by the free action of a
finite group. In this paper we study these threefolds under the assumption that
the group acts diagonally on the product. We show that the classification of
these threefolds is a finite problem, present an algorithm to classify them for
a fixed value of $\chi(\mathcal O_X)$ and explain a method to determine their
Hodge numbers. Running an implementation of the algorithm we achieve the full
classification of threefolds isogenous to a product of curves with
$\chi(\mathcal O_X)=-1$, under the assumption that the group acts faithfully on
each factor.
