A new paradigm views the recently launched Gravity Recovery and Climate Experiment (GRACE) follow-on mission as a gradiometer system, outside the mission's design concept, promising potentially major implications in geosciences. We develop an innovative and straightforward concept, the "GRACE gradiometer mode," to process future GRACE follow-on measurements to derive three-dimensional gravitational gradients. Using positions, accelerations, and attitude measurements from the identical predecessor mission GRACE, we generate common and differential accelerations. We validate GRACE differential accelerations using Gravity field and steady-state Ocean Circulation Explorer (GOCE) mission gradiometer measurements and we demonstrate an important and promising agreement between GRACE and GOCE. Consequently, we estimate gravitational gradients in full and we confirm their ability to detect geophysical signals over three example regions, namely, the Himalayas, Indonesia, and Canada. Coherence analysis between GOCE and GRACE gradients reveals a strong match that reaches up to 80%. GRACE gradiometer mode gradients are also compared with GRACE level 2- derived gradients, and results show that the new method captures smaller spatial-scale signals than GOCE with the trade-off being at higher noise level. We argue that future enhancements of the proposed proof-of-concept method will expand and enhance the GRACE follow-on objectives that will lead to new insights, discoveries, and applications in the Earth system and the geosciences.