A time domain integral equation (TDIE)-based approach for analyzing transient wave scattering from linear lossy media is proposed. The pertinent TDIEs are cast in terms of a conduction current corrected flux density and are solved using a marching-on-in-time (MOT) scheme that incorporates a differential equation update algorithm for the aforementioned flux. The scheme is accelerated by the PWTD algorithm, and it is shown that the computational complexity and memory requirements of the resulting solver scales as $cal O$(NtNs) and $cal O$(Ns), where Nt and Ns denote the number of temporal and spatial degrees of freedom in the flux expansion, respectively. Numerical results that validate the accuracy and efficacy of the proposed method are presented.