We show that applying the Laplace operator to a speckle-free quantitative phase image reveals an unprecedented level of detail in cell structure, without the gradient artifacts associated with differential interference contrast microscopy, or photobleaching and phototoxicity limitations common in fluorescence microscopy. This method, referred to as Laplace phase microscopy, is an efficient tool for tracking vesicles and organelles in living cells. The principle is demonstrated by tracking organelles in cardiomyocytes and vesicles in neurites of hippocampal neurons, which to our knowledge are the first label-free diffusion measurements of the organelles in such cells.