We introduce a novel block rational Krylov method to accelerate three-dimensional time-domain marine controlled-source electromagnetic modelling with multiple sources. This method approximates the time-varying electric solutions explicitly in terms of matrix exponential functions. A main attraction is that no time stepping is required, while most of the computational costs are concentrated in constructing a rational Krylov basis. We optimize the shift parameters defining the rational Krylov space with a positive exponential weight function, thereby producing smaller approximation errors at later times and reducing iteration numbers. Furthermore, for multi-source modelling problems, we adopt block Krylov techniques to incorporate all source vectors in a single approximation space. The method is tested on two examples: a layered seafloor model and a 3D hydrocarbon reservoir model with seafloor bathymetry. The modelling results are found to agree very well with those from 1D semi-analytic solutions and finite-element time-domain solutions using a backward Euler scheme, respectively. Numerical benchmarks show that the block method benefits from better memory and cache efficiency, resulting in about 1.26 to 1.48-fold speedup compared to non-block methods. Further efficiency gains are achieved through optimized rational Krylov techniques, allowing our approach to outperform classical time stepping schemes.