A hybrid approach, wherein Markov Chain Monte Carlo simulations are used in a Bayesian framework, in conjunction with artificial neural networks (ANN) is developed for solving an inverse heat conduction problem. Steady state three-dimensional heat conduction from a Teflon cylinder with uniform volumetric internal heat generation is considered. The goal is to estimate qv, given the heat transfer coefficient h, the thermal conductivity k and temperature data at certain fixed locations on the surface of the cylinder. For the purposes of establishing the soundness and efficacy of the approach, temperatures obtained by a numerical solution to the governing equation for known values of the parameter qv are first used to retrieve the quantities of interest, followed by retrievals with actual measurements. In order to significantly reduce the computational time associated with the MCMC simulations, first, a neural network is trained with limited number of solutions to the forward model. This serves as a surrogate to replace the forward model (conduction equation) during the process of retrievals with Markov Chain Monte Carlo simulations in a Bayesian framework. The performance of the proposed hybrid technique is evaluated for different cases.