Electromagnetism in an arbitrary gravitational background vacuum is formulated in terms of Clifford analysis over a pseudo-Riemannian space with signature (1,3). The full electromagnetic radiation problem is solved for given smooth compact support electric and magnetic monopole charge-current density source fields. We show that it is possible to attack this problem by analytical means in four dimensional form and without invoking electromagnetic potentials. Our approach reveals that the solution for the full electromagnetic field can be expressed in terms of a fundamental solution of the Laplace-de Rham scalar wave equation, so that calculating Green's dyadic fields is superfluous. In the absence of gravity our method reproduces (i) Jefimenko's equations and (ii) an expression for the particular solution of the wave equation satisfied by the electromagnetic field. Expression (ii) is simpler than Jefimenko's result and has the additional advantage that its evaluation avoids integrating over singularities.