The temporal stability of plane Poiseuille and Couette flows for a Navier-Stokes-Voigt type of viscoelastic fluid is investigated. The primary unidirectional flow is between two infinite rigid parallel plates, which are either fixed or in relative motion. The stability/instability of the basic flow is examined by carrying out a numerical solution of the stability eigenvalue problem. The stability characteristics are found to be different from that of Newtonian fluid despite the base flow remains the same. In the case of plane Poiseuille flow, two values of the Reynolds number are found to be needed to specify the linear instability criteria owing to the existence of closed neutral stability curves and also the instability emerges only in a certain range of Kelvin-Voigt parameter . To the contrary, instability occurs for all non-zero values of in the case of plane Couette flow and the sufficiency of a single critical value of the Reynolds number is ensued to discuss the stability/instability of fluid flow as the neutral stability curves are parabolic. The sensitivity of the Kelvin-Voigt parameter is clearly discerned on the stability of both types of flows. The variations of streamlines at the dominant mode of instability are analyzed to understand the instability mechanism and to provide information regarding the secondary flow pattern.