The shielding effectiveness of artificial periodic screens is investigated with reference to high-frequency high-impedance near fields produced by arbitrarily oriented electric dipoles. The screens are formed by metallic objects of arbitrary shape displaced in a 2-D lattice. The problem is first studied through a full-wave approach, using the array scanning method in conjunction with a periodic method of moments in the spatial domain. Next, the shielding problem is solved analytically in some characteristic electromagnetic compatibility configurations through the use of approximate low-frequency homogeneous models together with a classical analysis in the spectral domain. Finally, the solutions are compared with those deriving from the use of the so-called transmission-line approximation. The provided results show the suitability of the analytical approaches in dealing with finite sources different from standard plane-wave excitations and give a useful tool for the design of periodic shields.