The frequency following response (FFR) is a measure of the brain's periodic sound encoding. It is of increasing importance for studying the human auditory nervous system due to numerous associations with auditory cognition and dysfunction. Although the FFR is widely interpreted as originating from brainstem nuclei, a recent study using magnetoencephalography (MEG) suggested that there is also a right-lateralized contribution from the auditory cortex at the fundamental frequency (Coffey et al., 2016c). Our objectives in the present work were to validate and better localize this result using a completely different neuroimaging modality, and document the relationships between the FFR and the onset response, and cortical activity. Using a combination of electroencephalography, fMRI, and diffusion-weighted imaging, we show that activity in the right auditory cortex is related to individual differences in FFR-f0 strength, a finding that was replicated with two independent stimulus sets, with and without acoustic energy at the fundamental frequency. We demonstrate a dissociation between this FFR-f0-sensitive response in the right and an area in left auditory cortex that is sensitive to individual differences in the timing of initial response to sound onset. Relationships to timing and their lateralization are supported by parallels in the microstructure of the underlying white matter, implicating a mechanism involving neural conduction efficiency. These data confirm that the FFR has a cortical contribution, and suggest ways in which auditory neuroscience may be advanced by connecting early sound representation to measures of higher-level sound processing and cognitive function.