Phononic crystals (PnCs) are artificial composites designed by a periodic arrangement of two or more materials with different properties and/or geometry, in order to tailor the propagation of mechanical waves. The complex band structures of PnC thin plates, consisting of a hard matrix material reinforced by inclusions of soft viscoelastic material (HS-PnC), and vice-versa (SHPnC), are investigated. In general, elastic plate structures can be modelled using the Kirchhoff- Love theory, which is suitable for thin plates in low frequencies. However, there is a lack of knowledge concerning the effects of viscoelastic constituents on the complex band structure of flexural Bloch waves in PnC thin plates. In this study, an extended plane wave expansion (EPWE) method is used to compute the complex band structures of VPnC thin plates. The evanescent behaviour of periodic plates for the cases HS-PnC and SH-PnC is calculated, and the results show that the first creates flat bands for wave localization and the second opens up complete and partial Bragg scaterring band gaps.