Wire Arc Additive Manufacturing (WAAM) is a metal Additive Manufacturing (AM) technique that can produce fully dense metallic structures with virtually no porosity and at high productivity, compared to other currently available AM techniques such as Laser Powder Bed Fusion (L-PBF). As development of the technique is still ongoing, monitoring or post-fabrication inspection methods are under active investigation. In this work, we apply Resonant Ultrasound Spectroscopy (RUS) to samples fabricated from two different wires (construction steel and austenitic stainless steel) and quantitatively characterize isotropic and anisotropic elastic behaviour of the obtained dense parts. We find that an isotropic elastic model fits the construction steel samples well. For the 316 L polycrystal however, the isotropic elastic model is unsatisfactory, and an effective orthotropic elastic model is found to fit the resonance data. EBSD and XRD measurements are used to confirm and explain this difference in elastic behaviour between steel grades by the presence of a strong texture in the 316 L samples. Additionally, the texture data measured by EBSD are used to infer single crystal constants from the polycrystal resonance data using the Hill averaging scheme for one of the 316 L samples. We end by discussing the differences between the two elastic models used in the study (orthotropic and texture based) as well as the link between the measured resonances and microstructural descriptions of the samples.