The central objective of the present study is to generate an approximate, semi analytical piecewise solution of the one-dimensional, unsteady heat conduction equation in regular solid bodies (large wall, long cylinder and sphere) with uniform heat flux at the exposed surfaces. The main obstacle in these heat conduction problems lies in the boundary condition of uniform heat flux that is non-homogeneous, and of course is not conducive to the direct utilization of the method of separation of variables. To circumvent this restrictive obstacle, the traditional solution procedure relies on the principle of superposition of solutions composed of three degraded sub-solutions: one sub-solution in terms of the space coordinate and time, another sub-solution in terms of the space coordinate and another sub-solution in terms of time. In the present study, we exploit a radical different approach applying the Method Of Lines (MOL) forthwith to the one-dimensional, unsteady, heat conduction equation with prescribed uniform heat flux. Essentially, MOL discretizes the spatial partial derivatives in the one-dimensional, unsteady, heat conduction equation, while leaving the time partial derivative continuous. Correspondingly, the MOL transformation gives way to an adjoint system of linear, ordinary differential equations of first order in time, which is solved analytically (not numerically) with the potent eigenvalue method in the platform of a symbolic algebra code. The outcome of the two-part computational procedure provides a sequence of semi-analytical piecewise temperature distributions at the pre-selected lines in the large wall, long cylinder and sphere, which are expressed in terms of linear combinations of exponential functions of time comprising the required set of eigenvalues and eigenvectors. At the end, it was determined that the semi-analytical piecewise surface, center and mean temperature-time distributions based on seven lines exhibit excellent quality in the entire time domain.