Cortical bone is an anisotropic material with hierarchical structure whose elastic properties are described at different length scales. At millimeter-scale, the whole set of stiffness tensor can be conveniently measured by resonant ultrasound spectroscopy (RUS). At microscopic-scale, micro-indentation and scanning acoustic microscopy are often used to quantify tissue matrix elastic properties. However, the last two methods do not easily provide the entire tissue stiffness tensor, but rather the elasticity or stiffness along a single direction. To overcome this limitation, we introduce a new approach to retrieving the elastic tensor of the tissue based on a numerical optimization procedure using Fast Fourier Transform (FFT)-based homogenization for the forward computation.