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Spatiotemporal vortex pulses are wave packets that carry transverse orbital angular momentum, exhibiting exotic structured wave fronts that can twist through space and time. Existing methods to generate these pulses require complex setups like spatial light modulators or computer-optimized structures. Here, we demonstrate a new approach to generate spatiotemporal vortex pulses using just a simple diffractive grating. The key is constructing a phase vortex in frequency-momentum space by leveraging symmetry, resonance, and diffraction. Our approach is applicable to any wave system. We use a liquid surface wave (gravity wave) platform to directly demonstrate and observe the real-time generation and evolution of spatiotemporal vortex pulses. This straightforward technique provides opportunities to explore pulse dynamics and potential applications across different disciplines.