A quick and accurate method of measuring elastic and viscoelastic constants of a material is the essential first step for characterizing the material. This is more challenging for composite materials because unlike homogeneous metals and ceramics the material properties change from specimen to specimen for composite materials as the volume fraction of fibers and their orientations change. Anisotropic properties of composite materials add another difficulty in the measurement technique, since anisotropy increases the number of independent material constants. Polymer composites exhibit a high degree of attenuation in the matrix material; as a result, these composite materials cannot be assumed to be pure elastic material, so they should be modeled as viscoelastic materials by making the material constants complex. The real part is associated with the elastic behavior and the imaginary part is associated with the viscoelastic or attenuative behavior of the material. The number of independent material constants for a unidirectional (UD) composite, which is transversely isotropic, is ten (five real and five imaginary). Naturally, it is not practical and almost impossible to measure all these material constants by the traditional engineering method of applying stresses and measuring strains in different directions. Because of the measurement difficulty the imaginary parts of the material constants are often ignored. However, it should be mentioned here that it is important to measure the imaginary components of material constants because porosity and microcracking in the matrix due to material fatigue and aging affect the attenuation more than the elastic properties. In other words, the imaginary components of the material constants are a better indicator of material aging compared to the real components. Hence, an efficient technique to measure both real and imaginary components of the material constants is warranted and developed in this paper.