In speech-production research, three-dimensional (3D) MRI of the upper airway has provided insights into vocal tract shaping and data for its modeling. Small movements of articulators can lead to large changes in the produced sound, therefore improving the resolution of these data sets, within the constraints of a sustained speech sound (6-12 s), is an important area for investigation. The purpose of the study is to provide a first application of compressed sensing (CS) to high-resolution 3D upper airway MRI using spatial finite difference as the sparsifying transform, and to experimentally determine the benefit of applying constraints on image phase. Estimates of image phase are incorporated into the CS reconstruction to improve the sparsity of the finite difference of the solution. In a retrospective subsampling experiment with no sound production, 5x and 4x were the highest acceleration factors that produced acceptable image quality when using a phase constraint and when not using a phase constraint, respectively. The prospective use of a 5x undersampled acquisition and phase-constrained CS reconstruction enabled 3D vocal tract MRI during sustained sound production of English consonants /s/, /integral/, /l/, and /r/ with 1.5 x 1.5 x 2.0 mm(3) spatial resolution and 7 s of scan time.