In this work, a normal-mode solution is presented for the three-dimensional problem of acoustic scattering from a penetrable horizontally layered cylindrical obstacle in a shallow-water waveguide. The ocean environment around the obstacle is considered horizontally stratified and the bottom is assumed to be rigid. In the general case of depth-dependent sound-speed and density profiles (ssdp) the total acoustic field is calculated numerically, while an analytic solution is obtained in the special case of depth-independent ssdp. Numerical results concerning the transmission loss outside and inside single or double-layered cylindrical structures made of acoustic materials are given for a typical depth-dependent ocean environment. Comparisons with the cases of ideally soft and ideally hard cylindrical obstacles [J. Acoust. Soc. Am. 100, 206-218 (1996)] are also made, illustrating the effect of acoustic properties of the obstacle. An important feature, which clearly emerges from the theoretical analysis and the numerical results, is the necessity of including evanescent modes in the calculations, in order to obtain physically meaningful and numerically accurate results. Furthermore, analytical expressions for the scattering cross section of a (penetrable or impenetrable) cylindrical obstacle are derived in terms of the expansion coefficients of the pressure field, and their behavior as frequency increases is numerically investigated. The solution presented in this paper, although addressing a special geometry, provides a means for handling strong discontinuities in both vertical and horizontal directions, and can serve as a benchmark solution to a problem for which no general numerical model exists, i.e., modeling acoustic scattering from a 3D obstacle in a 3D shallow-water waveguide.