The electromagnetic radiation of a charge moving in an infinite 3D structure made of parallel wires is considered. The periods of the structure are assumed to be small; therefore, it can be described by an effective permittivity tensor. The charge velocity is perpendicular to the wires. Analytical and numerical investigations are performed, and some unusual properties of the radiation are noted. It is shown that the radiation propagates along the wires and concentrates near certain rays behind the charge. The wave field does not vary with distance from the charge along these rays (if energy loss in the medium is negligible). The prospects for the use of the structure under consideration for diagnostics of bunches are noted.