We study the dynamics of elongated axisymmetric particles undergoing shear flow between two parallel planar walls, under creeping-flow conditions. Particles are modeled as linear chains of touching spheres and it is assumed that walls are separated by a distance comparable to particle length. The hydrodynamic interactions of the chains with the walls are evaluated using our Cartesian-representation algorithm [Bhattacharya, Physica A 356, 294-340 (2005b)]. We find that when particles are far from both walls in a weakly confined system, their trajectories are qualitatively similar to Jeffery orbits in unbounded space. In particular, the periods of the orbits and the evolution of the azimuthal angle in the flow-gradient plane are nearly independent of the initial orientation of the particle. For stronger confinements, however, (i.e., when the particle is close to one or both walls) a significant dependence of the angular evolution on the initial particle configuration is observed. The phases of particle trajectories in a confined dilute suspension subject to a sudden onset of shear flow are thus slowly randomized due to unequal trajectory periods, even in the absence of interparticle hydrodynamic interactions. Therefore, stress oscillations associated with initially coherent particle motions decay with time. The effect of near contact particle-wall interactions on the suspension behavior is also discussed.