This study was designed to measure the ultrasonic attenuation within phantoms and tissue samples over a broad bandwidth and at many angles of incidence with respect to intrinsic orientations in order to elucidate both the frequency and angular dependence of the attenuation coefficient. Significant angular dependence, or anisotropy, of the attenuation was observed in canine myocardium (maximum to minimum ratio: 2.2 to 1) and a tissue mimicking phantom of oriented graphite fibers in gelatin (max to min: 2 to 1). In control studies, insignificant anisotropy was observed in the attenuation in canine liver samples and phantoms with graphite powder suspended in gelatin. Comparisons of the magnitude of variations of the oriented‐fiber phantom to that predicted by a viscous relative motion model are presented.