Crust and upper mantle structure under Tanzania, East Africa, is investigated using broadband waveforms and phase travel times collected from regional events recorded by the 1994--1995 Tanzania Broadband Seismic Experiment. Broadband displacement waveforms from the mL 5.9 Rukwa graben earthquake (18 August 1994) are modeled using a wave number integration algorithm and a new phase time inversion technique to infer a very simple crustal and upper mantle model that can be used for waveform studies of other regional earthquakes in the area. Displacement amplitudes for all phases in the data are predicted quite well using source parameters determined from a previous study of teleseismic waveforms. The crust is parameterized by a linear gradient in velocity defined by surface and Moho P velocities, Poisson's ratio, and crustal thickness. This simple parameterization is sufficient to explain arrival times and waveshapes of the three-component waveforms recorded between 200 and 800 km at stations located on the craton and Mozambique belt. The S wave train consists of Sn, multiple Sn, and critically reflected S wave multiples in the crustal waveguide, in addition to the fundamental mode Rayleigh wave. High phase velocity, critical S wave reflections are seen to form the low group velocity Lg phase. Prominent sP depth phases are seen in the regional waveform data which can be used to infer accurate source depths for events in the region. The Tanzania craton and adjacent Mozambique belt crustal structure is simple and laterally homogeneous, showing few effects of rift disruption. High wave velocities in the uppermost mantle suggest little thermal modification from rifting processes of the East African Rift.