The transmission of acoustic waves through centimeter-scale elastic and viscoelastic two-dimensional silicone rubber/air phononic crystal structures is investigated theoretically and experimentally. We introduce a finite difference time domain method for two-dimensional elastic and viscoelastic composite structures. Elastic fluid-solid phononic crystals composed of a two-dimensional array of cylindrical air inclusions in a solid rubber matrix, as well as an array of rubber cylinders in an air matrix, are shown to behave similarly to fluid-fluid composite structures. These systems exhibit very wide band gaps in their transmission spectra that extend to frequencies in the audible range of the spectrum. This effect is associated with the very low value of the transverse speed of sound in rubber compared to that of the longitudinal polarization. The difference in transmission between elastic and viscoelastic rubber/air crystals results from attenuation of transmission over a very wide frequency range, leaving only narrow passing bands at very low frequencies. These phononic crystals demonstrate the practical design of elastic or viscoelastic solid rubber/air acoustic band gap sound barriers with small dimensions.