Using novel interferometric quantitative phase microscopy methods, we demonstrate that the surface integral of the optical phase associated with live cells is invariant to cell water content. Thus, we provide an entirely noninvasive method to measure the nonaqueous content or "dry mass" of living cells. Given the extremely high stability of the interferometric microscope and the femtogram sensitivity of the method to changes in cellular dry mass, this new technique is not only ideal for quantifying cell growth but also reveals spatially resolved cellular and subcellular dynamics of living cells over many decades in a temporal scale. Specifically, we present quantitative histograms of individual cell mass characterizing the hypertrophic effect of high glucose in a mesangial cell model. In addition, we show that in an epithelial cell model observed for long periods of time, the mean squared displacement data reveal specific information about cellular and subcellular dynamics at various characteristic length and time scales. Overall, this study shows that interferometeric quantitative phase microscopy represents a noninvasive optical assay for monitoring cell growth, characterizing cellular motility, and investigating the subcellular motions of living cells.