High-temperature series for the susceptibility of the Ising model with random fields are derived for hypercubic lattices at any dimensionality and for the FCC lattice. The coefficients in the expansion in K= beta J are finite polynomials in lambda =(h2)av/J2. The authors derive the series up to seventh order in K for both gaussian and +or-H distributions of the fields. Two different methods are used to generate the series, and exact relations are derived which relate some of the coefficients to those of the susceptibility series for the pure model. The Dlog Pade analysis of the series confirms that d=6 is the upper critical dimensionality. The results are consistent with the absence of a ferromagnetic phase for d=2, and they are not inconsistent with a possible transition for d=3.