Five melt-bearing polycrystalline olivine aggregates have been newly prepared by hot isostatic pressing and tested at high temperature and pressure with torsional forced-oscillation and microcreep methods. Cylindrical specimens, varying in average grain size from 7 to 52 ¿m, were annealed and then tested during slow staged cooling under 200 MPa pressure from maximum temperatures of 1240--1300¿C where they contained basaltic melt fractions ranging from ~0.0001 to 0.037. For temperatures ≥1000¿C, pronounced departures from elastic behavior are evident in strain energy dissipation Q-1 and associated dispersion of the shear modulus G. In marked contrast with the high-temperature viscoelastic behavior of melt-free materials, a broad dissipation peak is observed for each of the melt-bearing specimens - superimposed upon a melt-enhanced level of monotonically frequency- and temperature-dependent background dissipation. The oscillation period at which the peak is centered decreases systematically with increasing temperature. A global model comprising an Andrade-pseudoperiod background plus Gaussian peak accounts adequately for the variation of Q-1 with frequency, temperature, average grain size and melt fraction. In the following paper (Part II) a microstructural explanation for the observed viscoelastic behavior is sought and the global model is used to extrapolate the experimental data to the conditions of teleseismic wave propagation in the Earth's upper mantle.