The 3D approach was employed for investigations of the stability loss of the solid circular cylinder made from viscoelastic composite material. This approach is based on investigations of the evolution of the initial infinitesimal imperfections of the cylinder within the scope of 3D geometrically nonlinear field equations of the theory of viscoelasticity for anisotropic bodies. The numerical results of the critical forces and critical time are presented and discussed. To illustrate the importance of the results obtained using the 3D approach, these results are compared with the corresponding ones obtained by employing various approximate beam theories. The viscoelasticity properties of the cylinder's material are described by the fractional-exponential operator. The numerical results and their discussion are presented for the case where the cylinder is made of a uni-directional fibrous viscoelastic composite material. In particular, it is established that the difference between the critical times obtained by employing 3D and third order refined beam theories becomes more non-negligible if the values of the external compressive force are close to the critical compressive force which is obtained at t = infinity (t denotes a time).