Accurate mechanical properties are critical to the design and use of composite material structures. Due to the available processing methods, the properties of ceramic matrix materials are especially sensitive to the geometry of the component and how it is made. A method is presented by which the ply-level elastic properties of a composite material can be obtained for a common structure; a thick, laminated tube. The mechanical and thermal response of the tubes is modeled by a planar cylindrical elasticity solution. Properties are determined from surface strain measurements of a thick tube subject to axial, torsional, pressure, and thermal loads. All elastic properties (including thermal expansion coefficients) can be obtained except the out-of-plane shear moduli (G13, G23) which are not involved in the planar elasticity solution employed. The ply-level properties are estimated by inversion of the elasticity solution in terms of the global strain measurements. A Least Squares optimization approach is used for the inversion of the elasticity solution. Application of the method for a filament wound aluminum oxide-aluminum oxide tube is presented. Advantages and limitations of the method are identified.