Although the East African rift system formed in cratonic lithosphere above a large‐scale mantle upwelling, some sectors have voluminous magmatism, while others have isolated, small‐volume eruptive centers. We conduct teleseismic shear wave splitting analyses on data from 5 lake‐bottom seismometers and 67 land stations in the Tanganyika‐Rukwa‐Malawi rift zone, including the Rungwe Volcanic Province (RVP), and from 5 seismometers in the Kivu rift and Virunga Volcanic Province, to evaluate rift‐perpendicular strain, rift‐parallel melt intrusion, and regional flow models for seismic anisotropy patterns beneath the largely amagmatic Western rift. Observations from 684 SKS and 305 SKKS phases reveal consistent patterns. Within the Malawi rift south of the RVP, fast splitting directions are oriented northeast with average delays of ~1 s. Directions rotate to N‐S and NNW north of the volcanic province within the reactivated Mesozoic Rukwa and southern Tanganyika rifts. Delay times are largest (~1.25 s) within the Virunga Volcanic Province. Our work combined with earlier studies shows that SKS‐splitting is rift parallel within Western rift magmatic provinces, with a larger percentage of null measurements than in amagmatic areas. The spatial variations in direction and amount of splitting from our results and those of earlier Western rift studies suggest that mantle flow is deflected by the deeply rooted cratons. The resulting flow complexity, and likely stagnation beneath the Rungwe province, may explain the ca. 17 Myr of localized magmatism in the weakly stretched RVP, and it argues against interpretations of a uniform anisotropic layer caused by large‐scale asthenospheric flow or passive rifting.