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To gain a complete view on the unique spatial attenuation phenomenon arising from the periodicity and the material viscoelasticity of the two-dimensional phononic crystals with finite out-of-plane extension, this paper presents a theoretical study of the viscoelastic phononic crystal slabs to investigate the wave transmission characteristics in terms of the dispersion relations for the propagating, evanescent, and complex modes. A numerical approach based on the theory of three-dimensional elasticity is proposed to eliminate the restrictions of plane analysis and to obtain the complex band diagrams using the wave expansion functions combined with the finite discretization technique. This formulation allows one to perform the simulations associated with the nonlinear frequency-dependent viscoelastic property which is accurately described by a generalized Maxwell model in this work. Numerical results are presented to show the validity of the established model and to reveal the dissipation mechanisms inside and outside the band gaps as well as to investigate the effects induced by the material viscoelasticity on the attenuation diagrams. Furthermore, the influences of the out-of-plane extension, the geometrical size and the shape of the etcfhed holes are detailed discussed to obtain thorough understandings of such a finite solid viscoelastic periodic lattice which is indispensable to practical applications. [ABSTRACT FROM AUTHOR]