The mechanical properties of the cell nucleus are increasingly recognized as critical in many biological processes. The deformability of the nucleus determines the ability of immune and cancer cells to migrate through tissues and across endothelial cell layers, and changes to the mechanical properties of the nucleus can serve as novel biomarkers in processes such as cancer progression and stem cell differentiation. However, current techniques to measure the viscoelastic nuclear mechanical properties are often time consuming, limited to probing one cell at a time, or require expensive, highly specialized equipment. Furthermore, many current assays do not measure time-dependent properties, which are characteristic of viscoelastic materials. Here, we present an easy-to-use microfluidic device that applies the well-established approach of micropipette aspiration, adapted to measure many cells in parallel. The device design allows rapid loading and purging of cells for measurements, and minimizes clogging by large particles or clusters of cells. Combined with a semi-automated image analysis pipeline, the microfluidic device approach enables significantly increased experimental throughput. We validated the experimental platform by comparing computational models of the fluid mechanics in the device with experimental measurements of fluid flow. In addition, we conducted experiments on cells lacking the nuclear envelope protein lamin A/C and wild-type controls, which have well-characterized nuclear mechanical properties. Fitting time-dependent nuclear deformation data to power law and different viscoelastic models revealed that loss of lamin A/C significantly altered the elastic and viscous properties of the nucleus, resulting in substantially increased nuclear deformability. Lastly, to demonstrate the versatility of the devices, we characterized the viscoelastic nuclear mechanical properties in a variety of cell lines and experimental model systems, including human skin fibroblasts from an individual with a mutation in the lamin gene associated with dilated cardiomyopathy, healthy control fibroblasts, induced pluripotent stem cells (iPSCs), and human tumor cells. Taken together, these experiments demonstrate the ability of the microfluidic device and automated image analysis platform to provide robust, high throughput measurements of nuclear mechanical properties, including time-dependent elastic and viscous behavior, in a broad range of applications.