An efficient marching-on-in-time (MOT) scheme is presented for solving electric, magnetic, and combined field integral equations pertinent to the analysis of transient electromagnetic scattering from perfectly conducting surfaces residing in an unbounded homogenous medium. The proposed scheme is the extension of the frequency-domain adaptive integral/pre-corrected fast-Fourier transform (FFT) method to the time domain. Fields on the scatterer that are produced by space-time sources residing on its surface are computed: 1) by locally projecting, for each time step, all sources onto a uniform auxiliary grid that encases the scatterer; 2) by computing everywhere on this grid the transient fields produced by the resulting auxiliary sources via global, multilevel/blocked, space-time FFTs; 3) by locally interpolating these fields back onto the scatterer surface. As this procedure is inaccurate when source and observer points reside close to each other; and 4) near fields are computed classically, albeit (pre-)corrected, for errors introduced through the use of global FFTs. The proposed scheme has a computational complexity and memory requirement of O(NtNslog2Ns) and O(Ns32/) when applied to quasiplanar structures, and of O(NtNs32/log2Ns) and O(Ns2) when used to analyze scattering from general surfaces. Here, Ns and Nt denote the number of spatial and temporal degrees of freedom of the surface current density. These computational cost and memory requirements are contrasted to those of classical MOT solvers, which scale as O(NtNs2) and O(Ns2), respectively. A parallel implementation of the scheme on a distributed-memory computer cluster that uses the message-passing interface is described. Simulation results demonstrate the accuracy, efficiency, and the parallel performance of the implementation.