In this, the first of two papers, we obtain a simple analytic formula for the photodetachment cross section of H− in crossed electric and magnetic fields. The three-dimensional semiclassical approximation predicts oscillations in the spectrum and these oscillations are correlated with closed classical orbits. In the following paper [A. D. Peters and J. B. Delos, Phys. Rev. A 47, 3036 (1993)] we derive fully-quantum-mechanical formulas for the cross section in perpendicular electric and magnetic fields and show how these results can be reduced to the semiclassical results of this paper.