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In this article, exact solutions of unsteady oscillatory generalized Burgers’ fluid are proposed for three different cases using Fourier transform approach. The fluid is electrically conducting under the influence of uniform transverse magnetic field and passing through the porous medium. MHD flows are induced by imposed periodic pressure gradients with smaller oscillations. Closed form solutions are obtained using Fourier sine transform, and several existing results are recovered as limiting cases. Furthermore, effects of different fluid parameters on the velocity profile are studied graphically. Analysis reveals that magnetic field and porosity parameter increases the velocity profile in case of Oldroyd-B and generalized Burgers’ fluid. It is also observed that magnetic field has more prominent effect on Burgers’ fluid as compared to Oldroyd-B fluid, while porosity parameter showed noticeable effect on Oldroyd-B fluid as compared to Burgers’ fluid.