Bayesian inference with sequential Monte Carlo is used to quantify the orientation distribution function coefficients and to calculate the fully anisotropic elastic constants of additively manufactured specimens from only the experimentally-measured resonant frequencies. The parallelizable and open-source SMCPy Python package enabled Bayesian inference within this new modeling framework, resulting in an order of magnitude reduction of the computation time for an 8-core machine. Residual stress-induced shifts on the resonant frequencies were explicitly accounted for during the Bayesian inference, enabling the estimation of their effect on the resonant frequencies without a stress-relief heat treatment. Additively manufactured cobalt-nickel-base superalloy (SB-CoNi-10C) specimens were sectioned at multiple inclinations relative to the build direction and scanned with resonant ultrasound spectroscopy to demonstrate characterization of any arbitrarily textured cubic microstructure through the resonant frequencies. The orientation distribution function coefficients of the textured polycrystalline microstructure were estimated in tensorial form to calculate both the 2nd order Hashin-Shtrikman bounds and the self-consistent estimate of the elastic constants, enabling accurate determination of all 21 possible independent elastic constants through the convergence constraints of the texture. Pole figures generated directly from the calculated texture coefficients showed good agreement with experimentally measured textures.