Triaxial compression testing of pure, polycrystalline water ice I at conditions relevant to planetary interiors and near-surface environments (differential stresses 0.45 to 10 MPa, temperatures 200 to 250 K, confining pressure 50 MPa) reveals that a complex variety of rheologies and grain structures may exist for ice and that rheology of ice appears to depend strongly on the grain structures. The creep of polycrystalline ice I with average grain size of 0.25 mm and larger is consistent with previously published dislocation creep laws, which are now extended to strain rates as low as 2×10⁻⁸ s⁻¹. When ice I is reduced to very fine and uniform grain size by rapid pressure release from the ice II stability field, the rheology changes dramatically. At 200 and 220 K the rheology matches the grain-size-sensitive rheology measured by Goldsby and Kohlstedt [1997, this issue] at 1 atm. This finding dispels concerns that the Goldsby and Kohlstedt results were influenced by mechanisms such as microfracturing and cavitation, processes not expected to operate at elevated pressures in planetary interiors. At 233 K and above, grain growth causes the fine-grained ice to become more creep resistant. Scanning electron microscopy investigation of some of these deformed samples shows that grains have markedly coarsened and the strain hardening can be modeled by normal grain growth and the Goldsby and Kohlstedt rheology. Several samples also displayed very heterogeneous grain sizes and high aspect ratio grain shapes. Grain-size-sensitive creep and dislocation creep coincidentally contribute roughly equal amounts of strain rate at conditions of stress, temperature, and grain size that are typical of terrestrial and planetary settings, so modeling ice dynamics in these settings must include both mechanisms.