Numerical Fresnel diffraction is broadly used in optics and holography in particular. So far, it has been implemented using convolutional approaches, spatial convolutions, or the fast Fourier transform. We propose a new way, to our knowledge, of computing Fresnel diffraction using Gabor frames and chirplets. Contrary to previous techniques, the algorithm has linear-time complexity, does not exhibit aliasing, does not need zero padding, has no constraints on changing shift/resolution/pixel pitch between source and destination planes, and works at any propagation distance. We provide theoretical and numerical analyses, detail the algorithm, and report simulation results with an accelerated GPU implementation. This algorithm may serve as a basis for more flexible, faster, and memory-efficient computer-generated holography methods.