The linear stability of the flow of an Oldroyd-B fluid through a linear array of cylinders confined in a channel is analyzed by computing both the steady-state flows and the linear stability of these states by finite element analysis. The linear stability of two-dimensional base flows to three-dimensional perturbations is computed both by time integration of the linearized evolution equations for infinitesimal perturbations and by an iterative eigenvalue analysis based on the Amoldi method. These flows are unstable to three-dimensional perturbations at a critical value of the Weissenberg number that is in very good agreement with the experimental observations by Liu [Viscoelastic Flow of Polymer Solutions around Arrays of Cylinders: Comparison of Experiment and Theory, Ph.D. dissertation, Massachusetts Institute of Technology, Cambridge, MA, 1997] for flow of a polyisobutylene Boger fluid through a linear periodic array of cylinders. The wave number of the disturbance in the neutral or spanwise direction also is in good agreement with experimental measurements. For closely spaced cylinders, the mechanism for the calculated instability is similar to the mechanism proposed by Joo and Shaqfeh [J. Fluid Mech. 262 (1994) 27] for the non-axisymmetric instability observed in viscoelastic Couette flow, where perturbations to the shearing component of the velocity gradient interact with the polymer stresses in the base flow. However, when the cylinder spacing is increased, we discover a new mechanism for instability that involves the coupling between the elongational component of the velocity gradient and the stream-wise normal stress in the base state. In addition, the most unstable disturbance in the Couette geometry is over-stable (leading to time-periodic states) whereas the instability calculated for closely spaced cylinders grows monotonically with time. This apparent inconsistency is resolved by the observation that in a complex flow, the evolution of the perturbations to the base flow can be transient in a Lagrangian frame of reference, while steady in an Eulerian frame.