Metropolis Monte Carlo simulation is a powerful tool for studying the equilibrium properties of matter. In complex condensed-phase systems, however, it is difficult to design Monte Carlo moves that facilitate rapid sampling of uncorrelated configurations by virtue of being both non-local and likely to be accepted. Here, we introduce a new class of moves based on \emph{nonequilibrium} dynamics: candidate configurations are generated through a finite-time process in which a system is actively driven out of equilibrium, and accepted with criteria that preserve the equilibrium distribution. The acceptance criteria is similar to the Metropolis acceptance probability, but it is based on nonequilibrium work rather than the instantaneous energy difference. While generating finite-time switching trajectories incurs an additional cost, we demonstrate how driving some degrees of freedom while allowing others to evolve naturally can lead to large enhancements in acceptance probabilities, greatly reducing structural correlation times in simulations of dense solvated systems. Using nonequilibrium driven processes allows physical insight about part of a system to be translated into efficient proposals for the entire system, vastly expanding the repertoire of useful Monte Carlo proposals. The described theoretical framework is applicable to sampling from both a single thermodynamic state or a mixture of thermodynamic states, and allows both coordinates and thermodynamic parameters to be driven in nonequilibrium proposals.