A theoretical analysis of an unsteady magnetohydrodynamic free-convective flow of a viscous incompressible and electrically conducting fluid between two concentric vertical cylinders is carried out considering thermal boundary condition of the second kind at the outer surface of the inner cylinder. The governing equations of motion and energy are transformed into ordinary differential equations using the Laplace transform technique. The ordinary differential equations are then solved analytically and the Riemann-sum approximation method is used to invert the Laplace domain into time domain. A parametric study depicting the effect of the various parameters on the temperature, velocity, and their related quantities is conducted. On the outer surface of the inner cylinder, the skin friction is seen to decrease with the Hartmann number and increase with time. An opposite behavior is seen on the inner surface of the outer cylinder.