The problem of uniform flow past a linear periodic array, or grating, of circular cylinders is solved by a new transform technique recently devised by the author. The solutions are expressed as an integral representation dependent on a set of coefficients which are the solutions of an explicit and well-conditioned linear system that is easily solved. We also give new explicit approximation formulas for the so-called blockage coefficients that are accurate well beyond the dilute limit relevant to cylinders that are small relative to their spacing. Finally, we describe an alternative approach to the flow problems through conformal mapping theory and produce a transform representation of a useful conformal mapping from the disc-in-channel geometry to a concentric annulus. The latter result is expected to be useful in many other applications.