A recursive solution procedure is developed to analyze the acoustical scattering by multilayer concentric circular cylindrical scatterers. The procedure is based on multiple scattering in the single-scatterer methodology originally proposed by one of the authors. The solution for a scatterer having an arbitrary number of layers is solved by recursively using the solution for a dual-layer scatterer. The solution procedure and its implementation are extensively validated. Computational characteristics, including potential pitfalls and their avoidance, are explored. The solution procedure can be used to analyze scatterers with continuously varying properties in the radial direction by approximating such a scatterer as a multilayer system. A comprehensive numerical example shows an interesting phenomenon: when a planar wave impinges onto a cooling tube in air, instead of forming a shadow zone, a strongly collimated beam is formed behind the cooling tube. This phenomenon can be explained as due to the gradient-index acoustical lens that is created around the cooling tube, which bends the sound and collimates it in the forward direction.