Frequency-dependent diffusion MRI (dMRI) using oscillating gradient encoding and diffusional kurtosis imaging (DKI) techniques have been shown to provide additional insight into tissue microstructure compared to conventional dMRI. However, a technical challenge when combining these techniques is that the generation of the large b-values (≥2000 s/mm) required for DKI is difficult when using oscillating gradient diffusion encoding. While efficient encoding schemes can enable larger b-values by maximizing multiple gradient channels simultaneously, they do not have sufficient directions to enable the estimation of directional kurtosis parameters. Accordingly, we investigate a DKI fitting algorithm that combines axisymmetric DKI fitting, a prior that enforces the same axis of symmetry for all oscillating gradient frequencies, and spatial regularization, which together enable robust DKI fitting for a 10-direction scheme that offers double the b-value compared to traditional encoding schemes. Using data from mice (oscillating frequencies of 0, 60, and 120 Hz) and humans (0 Hz only), we first show that axisymmetric DKI fitting provides comparable or even slightly improved image quality as compared to kurtosis tensor fitting, and improved DKI map quality when using an efficient encoding scheme with averaging as compared to a traditional scheme with more encoding directions. We also demonstrate that enforcing consistent axes of symmetries across frequencies improves fitting quality, and spatial regularization during fitting preserves spatial features better than using Gaussian filtering prior to fitting, which is an oft-reported pre-processing step for DKI. Thus, the use of an efficient 10-direction scheme combined with the proposed DKI fitting algorithm provides robust maps of frequency-dependent directional kurtosis which may offer increased sensitivity to cytoarchitectural changes that occur at various cellular spatial scales over the course of healthy aging, and due to pathological alterations.