A new numerical technique is presented for the analysis of diffraction of horizontally and vertically polarised waves by an array of conducting cylinders with arbitrary elemental cross-section, commonly known as transmission gratings. The method, which is on the finite-element technique, is simple in nature and involves no iterative calculations. Numerical results are obtained for gratings of rectangular and circular elements for both polarisations at arbitrary angles of incidence. These results compare favourably with the experimental and theoretical data available in the literature. They further reveal some of the interesting properties of the transmission gratings. The proposed solution technique is also applicable to a large class of problems in electromagnetics and physics and therefore should be of considerable interest to engineers and physicists.