We investigate the linear stability of the non-separated viscoelastic creeping flow through a periodically constricted channel. Global linear stability analysis against two-dimensional perturbations is performed by using two methods: (i) direct evaluation of the eigenspectrum of the generalized eigenvalue problem and (ii) time integration of the linearized governing equations. The Oldroyd-B constitutive equation is used to model the viscoelastic stress. Spatial discretization is accomplished by spectral collocation by employing Chebyshev expansion on a staggered grid in the wall normal direction and Fourier expansion in the axial direction. A stationary mode of instability is observed with critical weissenberg number that decreases with increasing channel wall amplitude and frequency. The critical eigenfunction is localized near the walls and has large axial wavenumbers. An analysis has been performed to assess the different contributions arising from the viscoelastic stress and velocity gradients to the mechanical energy budget of the critical disturbance. This study revealed that the instability arises due to the coupling of the shear components of the base flow velocity gradient with the viscoelastic stress perturbations. (C) 2004 Elsevier B.V. All rights reserved.