Quantitative assessment of myocardial stiffness is crucial to understand and evaluate cardiac biomechanics and function. Despite the recent progresses of ultrasonic shear wave elastography, quantitative evaluation of myocardial stiffness still remains a challenge because of strong elastic anisotropy. In this paper we introduce a smart ultrasound approach for non-invasive real-time quantification of shear wave velocity (SWV) and elastic fractional anisotropy (FA) in locally transverse isotropic elastic medium such as the myocardium. The approach relies on a simultaneous multidirectional evaluation of the SWV without a prior knowledge of the fiber orientation. We demonstrated that it can quantify accurately SWV in the range of 1.5 to 6 m/s in transverse isotropic medium (FA < 0.7) using numerical simulations. Experimental validation was performed on calibrated phantoms and anisotropic ex vivo tissues. A mean absolute error of 0.22 m/s was found when compared to gold standard measurements. Finally, in vivo feasibility of myocardial anisotropic stiffness assessment was evaluated in four healthy volunteers on the antero-septo basal segment and on anterior free wall of the right ventricle (RV) in end-diastole. A mean longitudinal SWV of 1.08 ± 0.20 m/s was measured on the RV wall and 1.74 ± 0.51 m/s on the septal wall with a good intra-volunteer reproducibility (±0.18 m/s). This approach has the potential to become a clinical tool for the quantitative evaluation of myocardial stiffness and diastolic function.