The subject of the paper is the derivation and analysis of new multidimensional, high-resolution, finite volume evolution Galerkin (FVEG) schemes for systems of nonlinear hyperbolic conservation laws. Our approach couples a finite volume formulation with approximate evolution operators. The latter are constructed using the bicharacteristics of the multidimensional hyperbolic system, such that all of the infinitely many directions of wave propagation are taken into account. In particular, we propose a new FVEG-scheme, which is designed in such a way that for a linear wave equation system the approximate evolution operator calculates any one-dimensional planar wave exactly. This operator makes the FVEG-scheme stable up to a natural CFL limit of 1. Using the results obtained for the wave equation system a new approximate evolution operator for the linearised Euler equations is also derived. The integrals over the cell interfaces also need to be approximated with care; in this case our choice of Simpson's rule is guided by stability analysis of model problems. Second order resolution is obtained by means of a piecewise bilinear recovery. Numerical experiments confirm the accuracy and multidimensional behaviour of the new scheme.