We present a continuous-wave instrument for non-invasive diffuse optical imaging of the breast in a parallel-plate transmission geometry. The instrument measures continuous spectra in the wavelength range 650-1000 nm, with an intensity noise level <1.5% and a spatial sampling rate of 5 points/cm in the x- and y-directions. We collect the optical transmission at four locations, one collinear and three offset with respect to the illumination optical fiber, to recover the depth of optical inhomogeneities in the tissue. We imaged a tissue-like, breast shaped, silicone phantom (6 cm thick) with two embedded absorbing structures: a black circle (1.7 cm in diameter) and a black stripe (3 mm wide), designed to mimic a tumor and a blood vessel, respectively. The use of a spatially multiplexed detection scheme allows for the generation of on-axis and off-axis projection images simultaneously, as opposed to requiring multiple scans, thus decreasing scan-time and motion artifacts. This technique localizes detected inhomogeneities in 3D and accurately assigns their depth to within 1 mm in the ideal conditions of otherwise homogeneous tissue-like phantoms. We also measured induced hemodynamic changes in the breast of a healthy human subject at a selected location (no scanning). We applied a cyclic, arterial blood pressure perturbation by alternating inflation (to a pressure of 200 mmHg) and deflation of a pneumatic cuff around the subject's thigh at a frequency of 0.05 Hz, and measured oscillations with amplitudes up to 1 μM and 0.2 μM in the tissue concentrations of oxyhemoglobin and deoxyhemoglobin, respectively. These hemodynamic oscillations provide information about the vascular structure and functional integrity in tissue, and may be used to assess healthy or abnormal perfusion in a clinical setting.