The problem of flickering trajectories in standard kinetic Monte Carlo (kMC) simulations prohibits sampling of the transition path ensembles (TPEs) on Markovian networks representing many slow dynamical processes of interest. In the present contribution, we overcome this problem using knowledge of the metastable macrostates, determined by an unsupervised community detection algorithm, to perform enhanced sampling kMC simulations. We implement two accelerated kMC methods to simulate the nonequilibrium stochastic dynamics on arbitrary Markovian networks, namely, weighted ensemble (WE) sampling and kinetic path sampling (kPS). WE-kMC utilizes resampling in pathway space to maintain an ensemble of representative trajectories covering the state space, and kPS utilizes graph transformation to simplify the description of an escape trajectory from a trapping energy basin. Both methods sample individual trajectories governed by the linear master equation with the correct statistical frequency. We demonstrate that they allow for efficient estimation of the time-dependent occupation probability distributions for the metastable macrostates, and of TPE statistics, such as committor functions and first passage time distributions. kPS is particularly attractive, since its efficiency is essentially independent of the degree of metastability, and we suggest how the algorithm could be coupled with other enhanced sampling methodologies. We illustrate our approach with results for a network representing the folding transition of a tryptophan zipper peptide, which exhibits a separation of characteristic timescales. We highlight some salient features of the dynamics, most notably, strong deviations from two-state behavior, and the existence of multiple competing mechanisms.