We present a novel third order fully-discrete Active Flux method based on the multidimensional approximate evolution operators. The latter are derived applying the theory of bicharacteristics. An approximate evolution operator is used to evolve edge values as independent variables (and thus the fluxes), thereby effectively doubling the degrees of freedom. However, the stability imposes stringent requirements on the CFL criterion. In order to improve the stability, we investigate the influence of different approximate evolution operators on the stability and accuracy of the resulting Active Flux method when applied to the linear acoustics system. In particular, we discuss the influence of various third-order reconstructions in space on the stability and accuracy of the Active Flux method. Further, we propose our recent results on the convergence analysis via dissipative measure-valued solutions and weak-strong uniqueness principle. Consequently, we show that the first order variant of the Active Flux finite volume method converges strongly to a strong solution of the multidimensional Euler system as long as the latter exists.