The linear stability and nonlinear dynamics of viscoelastic liquid films flowing down inclined surfaces with sinusoidal topography are investigated. The Oldroyd-B constitutive model is used and numerical solutions of a long-wave nonlinear evolution equation for the film thickness, introduced by Dávalos-Orozco [L. A. Dávalos-Orozco, Stability of thin viscoelastic films falling down wavy walls, Interfacial Phenom. Heat Transfer 1, 301 (2013)], provide insight into the influence of elasticity and wall topography on the nonlinear film dynamics, while Floquet analysis of the linearized evolution equation is used to study the onset of linear instability. Focusing initially on inertialess films (with zero Reynolds number), linear stability results are organized into three regimes based on the wall wavelength. For sufficiently short and sufficiently long wall wavelengths, the onset of instability is not tangibly affected by the topography. There is however an intermediate range of wavelengths where, as the wall wavelength is increased, the critical Deborah number for the onset of instability first decreases (topography is destabilizing) and then increases sufficiently for topography to be stabilizing (relative to the flat wall). Solutions to a perturbation amplitude equation indicate that the character of the instability changes substantially within this intermediate range; topography induces streamwise variations in the base-state velocity at the free surface which couple with perturbations and substantially influence the instability growth rate. Very similar trends are observed for Newtonian films and variations in the critical Reynolds number. Simulations of the full nonlinear evolution equation produce a broad range of nonlinear states including traveling waves, time-periodic waves, and chaos. Perturbations to the film generally saturate at higher amplitudes for cases with larger linear growth rates, e.g., with increasing Deborah number or for a destabilizing wall wavelength, and topography introduces finer temporal scales in the dynamics. The qualitative influences of inclination and inertia on the nonlinear dynamics are shown to be simply related to the influence of elasticity using analytical linear stability results for the flat-wall case.