It is well known that pseudospectral space discretizations of linear hyperbolic problems with periodic boundary conditions possess an infinite order of accuracy. These semidiscretizations are usually integrated in time by means of standard finite-difference procedures (which clearly are of finite order of accuracy), thus resulting in an unbalanced overall scheme. In this paper an explicit time-integration technique is employed which achieves infinite order of accuracy. The underlying idea, first suggested by {\it W. J. Cody, G. Meinardus} and {\it R. S. Varga} [J. Approx. Theory 2, 50-65 (1969;

Zbl 0187.116)] in the field of parabolic problems, is to approximate the exponential exp(tA) by a high degree minimax polynomial P(tA) rather than by a power $Q(\Delta tA)\sp{t/\Delta t}$, with Q($\Delta$ tA) a Taylor polynomial of exp($\Delta$ tA). Resolution, stability and accuracy are discussed. The paper includes numerical results for several scalar model problems.