[en] The Adjacent-cell Preconditioner (AP) is derived for accelerating generic fixed-weight, Weighted Diamond Difference (WDD) neutron transport methods in multidimensional Cartesian geometry. The AP is determined by requiring: (a) the eigenvalue of the combined mesh sweep-AP iterations to vanish in the vicinity of the origin in Fourier space; and (b) the diagonal and off-diagonal elements of the preconditioner to satisfy a diffusion-like condition. The spectra of the resulting iterations for a wide range of problem parameters exhibit a spectral radius smaller than .25, that vanishes implying immediate convergence for very large computational cells. More importantly, unlike other unconditionally stable acceleration schemes, the AP is cell-centered and its spectral radius remains small when the cell aspect ratio approaches 0 or ∞. Testing of the AP and comparison of its rate of convergence to the standard Source Iterations (SI) for Burre's Suite of Test Problems (BSTeP) demonstrates its high efficiency in reducing the number of iterations required to achieve convergence, especially for optically thick cells where acceleration is most needed