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We present time-resolved spectrophotometry of the pulsating DA white dwarf G29-38. As in previous broad-band photometry, the light curve shows the presence of a large number of periodicities. Many of these are combination frequencies, i.e. periodicities occurring at frequencies that are sums or differences of frequencies of stronger, real modes. We identify at least six real modes, and at least five combination frequencies. We measure line-of-sight velocities for our spectra and detect periodic variations at the frequencies of five of the six real modes, with amplitudes of up to 5km.s^–1. We argue that these variations reflect the horizontal surface motion associated with the g-mode pulsations. No velocity signals are detected at any of the combination frequencies, confirming that the flux variations at these frequencies do not reflect physical pulsation, but rather reflect mixing of frequencies owing to a non-linear transformation in the outer layers of the star. We discuss the amplitude ratios and phase differences found for the velocity and light variations, as well as those found for the real modes and their combination frequencies, both in a model-independent way and in the context of models based on the convective-driving mechanism. In a companion paper, we use the wavelength dependence of the amplitudes of the modes to infer their spherical degree.