Recent breakthroughs of cell phenotype reprogramming impose theoretical challenges on unravelling the complexity of large circuits maintaining cell phenotypes coupled at many different epigenetic and gene regulation levels, and quantitatively describing the phenotypic transition dynamics. A popular picture proposed by Waddington views cell differentiation as a ball sliding down a landscape with valleys corresponding to different cell types separated by ridges. Based on theories of dynamical systems, we establish a novel 'epigenetic state network' framework that captures the global architecture of cell phenotypes, which allows us to translate the metaphorical low-dimensional Waddington epigenetic landscape concept into a simple-yet-predictive rigorous mathematical framework of cell phenotypic transitions. Specifically, we simplify a high-dimensional epigenetic landscape into a collection of discrete states corresponding to stable cell phenotypes connected by optimal transition pathways among them. We then apply the approach to the phenotypic transition processes among fibroblasts (FBs), pluripotent stem cells (PSCs) and cardiomyocytes (CMs). The epigenetic state network for this case predicts three major transition pathways connecting FBs and CMs. One goes by way of PSCs. The other two pathways involve transdifferentiation either indirectly through cardiac progenitor cells or directly from FB to CM. The predicted pathways and multiple intermediate states are supported by existing microarray data and other experiments. Our approach provides a theoretical framework for studying cell phenotypic transitions. Future studies at single-cell levels can directly test the model predictions.