Characterizing the neural dynamics underlying sensory processing is one of the central areas of investigation in systems and cognitive neuroscience. Neuroimaging techniques such as magnetoencephalography (MEG) and Electroencephalography (EEG) have provided significant insights into the neural processing of continuous stimuli, such as speech, thanks to their high temporal resolution. Existing work in the context of auditory processing suggests that certain features of speech, such as the acoustic envelope, can be used as reliable linear predictors of the neural response manifested in MEG/EEG. The corresponding linear filters are referred to as temporal response functions (TRFs). While the functional roles of specific components of the TRF are well-studied and linked to behavioral attributes such as attention, the cortical origins of the underlying neural processes are not as well understood. Existing methods for obtaining the cortical representation of such linear speech processing work in a two-stage fashion: either the TRFs are first estimated at the sensor level, and then mapped to the cortex via source localization, or the neuroimaging data are first mapped to the cortex followed by estimating TRFs for each of the resulting cortical sources. Given that each stage is biased towards specific requirements, such as sparsity and smoothness, the end result typically suffers from destructive propagation of biases across stages, which in turn hinders a valid statistical interpretation of the results and requires significant post-hoc processing to summarize the results in a meaningful fashion. In this work we address this issue by estimating a linear filter representation of cortical sources directly from neuroimaging data. To this end, we introduce Neuro-Current Response Functions (NCRFs), which are three-dimensionally oriented linear filters spatially distributed throughout the cortex, that predict the cortical current dipoles giving rise to the observed MEG (or EEG) data in response to speech. NCRF estimation is cast within a Bayesian framework, which allows unification of the TRF and source estimation problems, and also facilitates the incorporation of prior information on the structural properties of the NCRFs. We present a fast estimation algorithm, which we refer to as the Champ-Lasso algorithm, by leveraging recent advances in optimization. We demonstrate the utility of the Champ-Lasso algorithm through application to simulated and experimentally recorded MEG data under auditory experiments. Our simulation studies reveal significant improvements over existing work, in terms of both spatial resolution and reliance on fine-tuned coordinate co-registration. Application to experimentally-recorded data corroborates existing results, while also delineating the distinct cortical distribution of the underlying neural processes at high spatiotemporal resolution and obviating the need for post-processing steps such as clustering and denoising. In summary, we provide a principled modeling and estimation paradigm for ME...