  Group lattices (Cayley digraphs) of a discrete group are in natural
correspondence with differential calculi on the group. On such a differential
calculus geometric structures can be introduced following general recipes of
noncommutative differential geometry. Despite of the non-commutativity between
functions and (generalized) differential forms, for the subclass of
``bicovariant'' group lattices considered in this work it is possible to
understand central geometric objects like metric, torsion and curvature as
``tensors'' with (left) covariance properties. This ensures that tensor
components (with respect to a basis of the space of 1-forms) transform in the
familiar homogeneous way under a change of basis. There is a natural
compatibility condition for a metric and a linear connection. The resulting
(pseudo-) Riemannian geometry is explored in this work. It is demonstrated that
the components of the metric are indeed able to properly describe properties of
discrete geometries like lengths and angles. A simple geometric understanding
in particular of torsion and curvature is achieved. The formalism has much in
common with lattice gauge theory. For example, the Riemannian curvature is
determined by parallel transport of vectors around a plaquette (which
corresponds to a biangle, a triangle or a quadrangle).
