We combine the transient thermal grating and time-domain thermoreflectance techniques to characterize the anisotropic thermal conductivities of GaAs/AlAs superlattices from the same wafer. The transient grating technique is sensitive only to the in-plane thermal conductivity, while time-domain thermoreflectance is sensitive to the thermal conductivity in the cross-plane direction, making them a powerful combination to address the challenges associated with characterizing anisotropic heat conduction in thin films. We compare the experimental results from the GaAs/AlAs superlattices with first-principles calculations and previous measurements of Si/Ge SLs. The measured anisotropy is smaller than that of Si/Ge SLs, consistent with both the mass-mismatch picture of interface scattering and with the results of calculations from density-functional perturbation theory with interface mixing included.