An important class of canonical problems which is employed in quantifying the slip-periness of microstructured superhydrophobic surfaces is concerned with the calculationof the hydrodynamic loads on adjacent solid bodies whose size is large relative to themicrostructure period. The effect of superhydophobicity is most pronounced when thelatter period is comparable with the separation between the solid probe and the su-perhydrophobic surface. We address the above distinguished limit, considering a simpleconfiguration where the superhydrophobic surface is formed by a periodically groovedarray, in which air bubbles are trapped in a Cassie state, and the solid body is an in-finite cylinder. In the present Part, we consider the case where the grooves are alignedperpendicular to the cylinder and allow for three modes of rigid-body motion: rectilinearmotion perpendicular to the surface; rectilinear motion parallel to the surface, in thegrooves direction; and angular rotation about the cylinder axis. In this scenario, the flowis periodic in the direction parallel to the axis. Averaging over the small-scale periodicityyields a modified lubrication description where the small-scale details are encapsulatedin two auxiliary two-dimensional cell problems which respectively describe pressure- andboundary-driven longitudinal flow through an asymmetric rectangular domain, boundedby a compound surface from the bottom and a no-slip surface from the top. Once theintegral flux and averaged shear stress associated with each of these cell problems arecalculated as a function of the slowly varying cell geometry, the hydrodynamic loadsexperienced by the cylinder are provided as quadratures of nonlinear functions of thelatter distributions over a continuous sequence of cells.