High-fidelity fluid simulations are central to understanding complex transport phenomena yet resolving large or geometrically intricate systems remains computationally prohibitive due to the massive memory requirements of Direct Numerical Simulation (DNS). In this presentation, we introduce MPS-LBM, a novel quantum-inspired solver that integrates the Lattice Boltzmann Method (LBM) with Matrix Product States (MPS) to address these scalability bottlenecks. While previous tensor-network approaches have been largely restricted to simple domains, we extend the applicability of MPS techniques to three-dimensional flows characterized by complex boundaries and structured media. Our method decomposes the LBM particle distribution functions into MPS form, enabling efficient collision steps to be performed via element-wise operations and streaming to be executed as exact low-rank Matrix Product Operators (MPO) within the compressed manifold. Furthermore, we introduce a robust approach to handling complex geometries through MPS-decomposed binary masks, enabling the simulation of arbitrary boundary conditions. We validate the accuracy and efficiency of MPS-LBM across three distinct 3D test cases: the Taylor-Green vortex, blood flow through an aneurysm, and flow through a pin-fin heat exchanger array. A key insight of our work is that flows within structured domains possessing translational symmetries exhibit very high compression rates. Consequently, we demonstrate that MPS-LBM can achieve compression ratios exceeding two orders of magnitude in pin-fin configurations while maintaining high dynamical fidelity and physical structure. These results position tensor networks as a scalable paradigm for computational fluid dynamics, enabling the systematic exploration of regimes previously intractable with classical methods.