With multiple possible mediators on the causal pathway from a treatment to an outcome, we consider the problem of decomposing the effects along multiple possible causal path(s) through each distinct mediator. Under a path-specific effects framework, such fine-grained decompositions necessitate stringent assumptions, such as correctly specifying the causal structure among the mediators, and no unobserved confounding among the mediators. In contrast, interventional direct and indirect effects for multiple mediators can be identified under much weaker conditions, while providing scientifically relevant causal interpretations. Nonetheless, current estimation approaches require (correctly) specifying a model for the joint mediator distribution, which can be difficult when there is a high-dimensional set of possibly continuous and noncontinuous mediators. In this article, we avoid the need to model this distribution, by developing a definition of interventional effects previously suggested for longitudinal mediation. We propose a novel estimation strategy that uses nonparametric estimates of the (counterfactual) mediator distributions. Noncontinuous outcomes can be accommodated using nonlinear outcome models. Estimation proceeds via Monte Carlo integration. The procedure is illustrated using publicly available genomic data to assess the causal effect of a microRNA expression on the 3-month mortality of brain cancer patients that is potentially mediated by expression values of multiple genes.