© 2020The technical paper addresses unsteady heat conduction in large, square crossbars and cubes heated continually with uniformly surface heat flux. The uniform surface heat flux is customarily provided by electrical, radiative and pool fire heating in engineering applications. This type of heating fall under the category of non-homogeneous Neumann boundary conditions. The existing traditional method for solving the unsteady, heat conduction equation in two and three dimensions with constant thermo-physical properties and subject to non-homogeneous uniform heat flux boundary conditions is the principle of superposition of partial solutions of simpler unsteady, heat conduction equations. For the latter, the exact, analytical temperature distributions are customarily expressed by infinite series regardless of the solution method. For the numerical evaluation of spatio-temporal temperatures, the main inconvenience of the infinite series lies in the poor convergence when the time is small. To prevent the impeding obstacle, the Transversal Method Of Lines (TMOL) is implemented in this work to transform the unsteady, two- and three-dimensional heat conduction equations into degraded “quasi steady”, two- and three-dimensional heat conduction equations, namely the homogeneous Helmhotz equations. Thereby, the exact, analytical solutions of the converted partial differential equations of elliptic type and homogeneous were determined by the standard method of separation of variables. The resulting approximate, semi-analytical two- and three-dimensional temperature distributions for large, square crossbars and cubes do not contain elaborate infinite series. Instead, the temperature disdtributions contain hyperbolic functions elucidating worthy compactness and superb quality, regardless of the time, either “small time”, “moderate time” or “large time”.