The lattice Boltzmann method often involves small numerical time steps due to the acoustic scaling (i.e., scaling between time step and grid size) inherent to the method. In this work, a second-order dual-time-stepping lattice Boltzmann method is proposed in order to avoid any time-step restriction. The implementation of the dual time stepping is based on an external source in the lattice Boltzmann equation, related to the time derivatives of the macroscopic flow quantities. Each time step is treated as a pseudosteady problem. The convergence rate of the steady lattice Boltzmann solver is improved by implementing a multigrid method. The developed solver is based on a two-relaxation time model coupled to an immersed-boundary method. The reliability of the method is demonstrated for steady and unsteady laminar flows past a circular cylinder, either fixed or towed in the computational domain. In the steady-flow case, the multigrid method drastically increases the convergence rate of the lattice Boltzmann method. The dual-time-stepping method is able to accurately reproduce the unsteady flows. The physical time step can be freely adjusted; its effect on the simulation cost is linear, while its impact on the accuracy follows a second-order trend. Two major advantages arise from this feature. (i) Simulation speed-up can be achieved by increasing the time step while conserving a reasonable accuracy. A speed-up of 4 is achieved for the unsteady flow past a fixed cylinder, and higher speed-ups are expected for configurations involving slower flow variations. Significant additional speed-up can also be achieved by accelerating transients. (ii) The choice of the time step allows us to alter the range of simulated timescales. In particular, increasing the time step results in the filtering of undesired pressure waves induced by sharp geometries or rapid temporal variations, without altering the main flow dynamics. These features may be critical to improve the efficiency and range of applicability of the lattice Boltzmann method.