Network oscillations are ubiquitous across many brain regions. In the basal ganglia, oscillations are also present at many levels and a wide range of characteristic frequencies have been reported to occur during both health and disease. The striatum, the main input nucleus of the basal ganglia, receives massive glutamatergic inputs from the cortex and is highly susceptible to external oscillations. However, there is limited knowledge about the exact nature of this routing process and therefore, it is of key importance to understand how time-dependent, external stimuli propagate through the striatal circuitry. Using a network model of the striatum and corticostriatal projections, we try to elucidate the importance of specific GABAergic neurons and their interactions in shaping striatal oscillatory activity. Here, we propose that fast-spiking interneurons can perform an important role in transferring cortical oscillations to the striatum especially to those medium spiny neurons that are not directly driven by the cortical oscillations. We show how the activity levels of different populations, the strengths of different inhibitory synapses, degree of outgoing projections of striatal cells, ongoing activity and synchronicity of inputs can influence network activity. These results suggest that the propagation of oscillatory inputs into the medium spiny neuron population is most efficient, if conveyed via the fast-spiking interneurons. Therefore, pharmaceuticals that target fast-spiking interneurons may provide a novel treatment for regaining the spectral characteristics of striatal activity that correspond to the healthy state.