This study is concerned with analytic solutions of some magnetohydrodynamic (MHD) flows of an Oldroyd-B fluid. The fluid is assumed to be electrically conducting in the presence of a uniform magnetic field and occupies the space over a flat plate between two sidewalls perpendicular to the plate. The governing equation of motion is treated analytically, yielding the derivation of exact solutions under integral and series form. By assigning numerical values to various parameters, the integrals encountered are evaluated numerically. In the absence of sidewalls, all solutions that have been obtained reduce to those corresponding to the motion over an infinite plate. The impacts of several quantities are investigated on the resulting velocity field and the tangential stress at the bottom wall.