Purpose: To present a computational procedure for accelerated, calibrationless magnetic resonance image (Cl-MRI) reconstruction that is fast, memory efficient, and scales to high-dimensional imaging. Theory and Methods: Cl-MRI methods can enable high acceleration rates and flexible sampling patterns, but their clinical application is limited by computational complexity and large memory footprint. The proposed computational procedure, HIgh-dimensional fast ConvolU-tional framework (HICU), provides fast, memory-efficient recovery of unsampled k-space points. For demonstration, HICU is applied to six 2D T2-weighted brain, seven 2D cardiac cine, five 3D knee, and one multi-shot diffusion weighted imaging (MSDWI) datasets. Results: The 2D imaging results show that HICU can offer one to two orders of magnitude computation speedup compared to other Cl-MRI methods without sacrificing imaging quality. The 2D cine and 3D imaging results show that the computational acceleration techniques included in HICU yield computing time on par with SENSE-based compressed sensing methods with up to 3 dB improvement in signal-to-error ratio and better perceptual quality. The MSDWI results demonstrate the feasibility of HICU for a challenging multi-shot echo-planar imaging application. Conclusions: The presented method, HICU, offers efficient computation and scalability as well as extendibility to a wide variety of MRI applications.