We show that a class of pyrochlore materials provides a route to realize 3D Quantum Hall insulators, based on magnetic texture and flat band interference. We study itinerant electrons on a pyrochlore lattice Kondo-coupled to localized moments living on another interpenetrating pyrochlore lattice, with the moments forming a field-selected three-in/one-out texture. The noncoplanar texture generates complex spin-dependent hopping amplitudes and unequal local exchange energies, reconstructing the compact-localized parent flat bands into narrow bands separated by a global three-dimensional gap. The low-energy states are concentrated predominantly on kagome planes within the three-dimensional lattice, reflecting the combined effects of quantum interference and the texture-induced sublattice potential. When the lowest band is fully occupied, the response is characterized by a Chern triplet C=(1,1,1) and reciprocal Hall vector G_H=(2π/a)(1,1,1), giving σ_yz=σ_zx=σ_xy=e²/(ha). The quantized bulk Hall response enforces chiral surface spectral flow, visible as noncontractible Fermi loops at the surface. Our work opens a new direction in the search for three-dimensional quantum Hall physics in magnetic pyrochlore systems, and suggests a broad design principle based on the interplay of magnetic texture and flat-band interference.