Dense cores are the simplest star-forming sites. They represent the end stage of the fragmentation hierarchy that characterizes molecular clouds, and they likely control the efficiency of star formation via their relatively low numbers. Recent dust continuum observations of entire molecular clouds show that dense cores often lie along large-scale filamentary structures, suggesting that the cores form by some type of fragmentation process in an approximately cylindrical geometry. To understand the formation mechanism of cores, additional kinematic information is needed, and this requires observations in molecular-line tracers of both the dense cores and their surrounding cloud material. Here I present some recent efforts to clarify the kinematic structure of core-forming regions in the nearby Taurus molecular cloud. These new observations show that the filamentary structures seen in clouds are often more complex than suggested by the maps of continuum emission, and that they consist of multiple fiber-like components that have different velocities and sonic internal motions. These components likely arise from turbulent fragmentation of the large-scale flows that generate the filamentary structures. While not all these fiber-like components further fragment to form dense cores, a small group of them does so, likely by gravitational instability. This fragmentation produces characteristic chain-like groups of dense cores that further evolve to form stars.