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We report AC-impedance studies on a series of high-quality LiMn_1−xFe_xPO₄ single crystals with 0≤x≤1. Our results confirm quasi-one-dimensional transport in LiFePO₄ with fast Li-diffusion along the b-axis. The conductivities along the crystallographic b-, c- and a-axis differ by a factor of about 10, respectively. Whereas, the activation energy E_A of the effective diffusion process is particularly large for the b-axis and smallest for the a-axis. Remarkably, the b-axis ionic bulk conductivity of LiMn_0.5Fe_0.5PO₄ is of the same order of magnitude as in undoped LiFePO₄, which implies similarly fast Li-transport even upon 50% Mn-doping which, owing to the higher redox potential of the Mn³⁺/Mn²⁺-couple, yields enhanced energy density in lithium-ion batteries. The overall results of our impedance studies draw a far more complex picture than it would be expected from a simple one-dimensional ionic conductor. Our results suggest a strong contribution of crystal defects in actual materials.