We study scalar, fermionic and gauge fields coupled nonminimally to gravity in the Einstein-Cartan formulation. We construct a wide class of models with nondynamical torsion whose gravitational spectra comprise only themassless graviton. Eliminating nonpropagating degrees of freedom, we derive an equivalent theory in themetric formulation of gravity. It features contact interactions of a certain formbetween, and among, the matter and gauge currents. We also discuss briefly the inclusion of curvature-squared terms.