We study the continuum limit for Dirac-Hodge operators defined on the $n$-dimensional square lattice $h\mathbb{Z}^n$ as $h$ goes to $0$. This result extends, to a first order discrete differential operator, the known convergence of discrete Schrödinger operators to their continuous counterpart.To establish this discrete analog, we introduce an alternative framework for higher-dimensional discrete differential calculus compared to the standard one defined on simplicial complexes. Subsequently, we express our operator as a differential operator acting on discrete forms, enabling us to demonstrate the convergence to the continuous Dirac-Hodge operator.