A method to measure phase velocity and attenuation of compressional and shear waves has been developed using a broadband through-transmission technique. Two pairs of transducers were used, one pair with center frequency 5 MHz (Olympus C309-SU), and the other pair with center frequency 10 MHz (Olympus V327-SU). The method was tested on the homogeneous material Polymethylmethacrylate (PMMA) and the composite material Eccosorb MF-117. Our results were compared with published values for the PMMA material, showing a good agreement with literature, hence, verifying the accuracy of our method. In addition, the temperature in the sample was varied from 19 °C to 37 °C, to investigate how this influences the parameters. A small variation of around 3% was found for the phase velocity, while the temperature influence on the attenuation was considerably larger, increasing 30% when temperature was changed from 19 °C to 37 °C. These acoustic properties and their temperature effects are essential for transducer designs, especially for clinical applications.