The elastic wave attenuation in artificial composites, known as phononic crystals (PnCs), is an important topic in the context of wave manipulation. However, there is a lack of knowledge in the obtainment of the wave attenuation of PnCs when the viscoelastic effect of different polymers is considered. In this study, the complex band structure of longitudinal waves in 1-D viscoelastic PnC (VPnC) rods composed by steel inclusions (elastic material) in a polymeric matrix (viscoelastic material) is investigated. The viscoelastic effect is modelled by the standard linear solid model (SLSM), which can be used to closely model the behavior of polymers for practical applications. It is also studied the influence of different polymers (i.e., epoxy, nylon, silicon rubber, natural rubber, and low density polyethylene (LDPE)) on the complex band structure of 1-D VPnC rods. The improved plane wave expansion (IPWE) and extended plane wave expansion (EPWE) are used to compute the band structure. It is observed that the viscoelastic effect influences significantly both the propagating and evanescent waves. The viscoelasticity increases the unit cell wave attenuation for most range of frequency considering all polymeric matrices. The highest unit cell wave attenuation is for the polymeric matrix of natural rubber.