Brain activation mapping using functional magnetic resonance imaging (fMRI) has been extensively studied in brain gray matter (GM), whereas in large disregarded for probing white matter (WM). This unbalanced treatment has been in part due to controversies in relation to the nature of the blood oxygenation level-dependent (BOLD) contrast in WM and its detachability. However, an accumulating body of studies has provided solid evidence of the functional significance of the BOLD signal in WM and has revealed that it exhibits anisotropic spatio-temporal correlations and structure-specific fluctuations concomitant with those of the cortical BOLD signal. In this work, we present an anisotropic spatial filtering scheme for smoothing WM fMRI data that accounts for known spatial constraints on the BOLD signal in WM. In particular, the spatial correlation structure of the WM BOLD signal is highly anisotropic and closely linked to local axonal structure in terms of shape and orientation, suggesting that isotropic Gaussian filters conventionally used for smoothing fMRI data are inadequate for denoising the WM BOLD signal as they are incapable of adapting to the underlying domain of the BOLD signal in white matter. The fundamental element in the proposed method is a graph-based description of the WM that encodes the underlying anisotropy observed across WM, derived from diffusion MRI data. Based on this representation, and leveraging graph signal processing principles, we design subject-specific spatial filters that adapt to a subject's unique WM structure at each position in the WM that they are applied at. We use the proposed filters to spatially smooth WM fMRI data, as an alternative to the conventional practice of using isotropic Gaussian filters. We test the proposed filtering approach on two sets of phantoms, showcasing its greater sensitivity and specificity for the detection of slender anisotropic activations, compared to that achieved with isotropic Gaussian filters. We also present WM activation mapping results on the Human Connectome Project's 100-unrelated subject dataset, across seven functional tasks, showing the capacity of the proposed method for detecting streamline-like activations within axonal bundles.