We formulate and sketch a solution method for the boundary value problem of electromagnetic radiation in a vacuum with a given gravitational background, in terms of Clifford Analysis over a pseudo-Riemannian manifold with signature (1, 3). The general solution for the electromagnetic field, generated by a compact smooth source, is obtained in a generalized integral representation form. Our method requires solving the scalar wave equation for a fundamental solution and solving the geodesic equation for local causalgeodesics. The considered problem is quite instructional for revealing the intricacies that arise when one tries to extend Clifford Analysis to the setting of pseudo- Riemannian manifolds. It is found, at least for the problem considered here, that such an extension is possible in principle, provided that several major technical difficulties are properly addressed. These difficulties are identified and for each of them a solution is proposed. © 2010 Springer Basel AG, Switzerland.