A critical consideration in tissue engineering is to recapitulate the microstructural organization of native tissues that is essential to their function. Scaffold-based techniques have focused on achieving this via the contact guidance principle wherein topographical cues offered by scaffold fibers direct migration and orientation of cells to govern subsequent cell-secreted extracellular matrix organization. Alternatively, approaches based on acoustophoretic, electrophoretic, photophoretic, magnetophoretic, and chemotactic principles are being investigated in the biofabrication domain to direct patterning of cells within bioink constructs. This work describes a new acoustophoretic three-dimensional (3D) biofabrication approach that utilizes radiation forces generated by superimposing ultrasonic bulk acoustic waves (BAW) to preferentially organize cellular arrays within single and multi-layered hydrogel constructs. Using multiphysics modeling and experimental design, we have characterized the effects of process parameters including ultrasound frequency (0.71, 1, 1.5, 2 MHz), signal voltage amplitude (100, 200 mVpp), bioink viscosity (5, 70 cP), and actuation duration (10, 20 min) on the alignment characteristics, viability and metabolic activity of human adipose-derived stem cells (hASC) suspended in alginate. Results show that the spacing between adjacent cellular arrays decreased with increasing frequency (p < 0.001), while the width of the arrays decreased with increasing frequency and amplitude (p < 0.05), and upon lowering the bioink viscosity (p < 0.01) or increasing actuation duration (p < 0.01). Corresponding to the computational results wherein estimated acoustic radiation forces demonstrated a linear relationship with amplitude and a nonlinear relationship with frequency, the interaction of moderate frequencies at high amplitudes resulted in viscous perturbations, ultimately affecting the hASC viability (p < 0.01). For each combination of frequency and amplitude at the extremities of the tested range, the hASC metabolic activity did not change over 4 d, but the activity of the low frequency-high amplitude treatment was lower than that of the high frequency-low amplitude treatment at day 4 (p < 0.01). In addition to this process-structure characterization, we have also demonstrated the 3D bioprinting of a multi-layered medial knee meniscus construct featuring physiologically-relevant circumferential organization of viable hASC. This work contributes to the advancement of scalable biomimetic tissue manufacturing science and technology.