The buckling instability of a thick rectangular plate made of a viscoelastic composite material is studied. The investigation is carried out within the framework of the three-dimensional linearized theory of stability. The plate edges are clamped and the plate is compressed through the clamps. Moreover, it is assumed that the plate has an initial infinitesimal imperfection, and, as a buckling criterion, the state is taken where this imperfection starts to increase indefinitely at fixed finite values of external compressive forces. From this criterion, the critical time is determined The corresponding boundary-value problems are solved by employing the three-dip mensional FEM and the Laplace transform. The material of the plate is assumed orthotropic, viscoelastic, and homogeneous. Numerical results related to the critical time are presented.