  We construct non-extremal as well as extremal black string solutions in
minimal five-dimensional supergravity coupled to vector multiplets using
dimensional reduction to three Euclidean dimensions. Our method does not assume
that the scalar manifold is a symmetric space, and applies as well to a class
of non-supersymmetric theories governed by a generalization of special real
geometry. We find that five-dimensional black string solutions correspond to
geodesics in a specific totally geodesic para-K\"ahler submanifold of the
scalar manifold of the dimensionally reduced theory, and identify the subset of
geodesics that corresponds to regular black string solutions in five
dimensions. BPS and non-BPS extremal solutions are distinguished by whether the
corresponding geodesics are along the eigendirections of the para-complex
structure or not, a characterization which carries over to non-supersymmetric
theories. For non-extremal black strings the values of the scalars at the outer
and inner horizon are not independent integration constants but determined by
certain functions of the charges and moduli. By lifting solutions from three to
four dimensions we obtain non-extremal versions of small black holes, and find
that while the outer horizon takes finite size, the inner horizon is still
degenerate.
