This study investigates an approach to determine the thermal conductivity of a material from transient temperature data. For this study, a one-dimensional transient convection heat diffusion PDE with a closed-form solution is used to model slender test coupons. The inhomogeneous boundary condition is handled using eigenfunction expansion and Green's second identity. A modified Levenberg-Marquardt (LM) nonlinear least squares (NLS) algorithm is used to determine the thermal conductivity from the experimental data using a flux boundary condition. The boundary conditions require an open-loop heater control and forced convection along the rod, which is simpler to implement than current methods. The estimated thermal conductivity from the experimental data was within ten percent of the nominal published values of the materials tested. The method described here promises to be easier to implement than current standards with similar accuracy and is applicable over a wide range of thermal conductivities -ranging from 15 to 400 W/mK. The method utilizes a small amount of material and a simple geometry, and is therefore suitable for the thermal characterization of additively-manufactured materials on a batch-by-batch basis.