A new approach to building explicit time-marching stencil computation schemes for the transient 2D acoustic wave equation is implemented. It is based on using Poisson’s formula and its three time level modification combined with polynomial stencil interpolation of the solution at each time-step and exact integration. The time-stepping algorithm consists of two explicit stencil computation procedures: a first time-step procedure incorporating the initial conditions and a two-step scheme for the second and next time-steps. Three particular explicit stencil schemes (with five, nine and 13 space points) are constructed using this approach. Their stability regions are presented. Accuracy advantages of the new schemes in comparison with conventional finite-difference schemes are demonstrated by simulation using an exact benchmark solution.