Three dimensional (3D) cell culture is becoming mainstream as it is recognized that many animal cell types require the biophysical and biochemical cues within the extracellular matrix (ECM) to perform truly physiologically realistic functions. However, tools for characterizing cellular mechanical environment are largely limited to cells cultured on a 2D substrate. We present a three dimensional (3D) traction microscopy that is capable of mapping 3D stress and strain within a soft and transparent ECM using a fluorescence microscope and a simple forward data analysis algorithm. We validated this technique by mapping the strain and stress field within the bulk of a thin polyacrylamide gel layer indented by a millimeter size glass ball, together with a finite element analysis. The experimentally measured stress and strain fields are in excellent agreements with results of the finite element simulation. The unique contributions of the presented 3D traction microscopy method are: (a) the use of a fluorescence microscope in contrast with the confocal microscope that is required for the current 3D traction microscopy in the literature; (b) the determination of the pressure field of an incompressible gel from strains; (c) the simple forward data analysis algorithm compatible with a nonlinear ECM. We apply our 3D traction microscopy method to map the 3D stress field over time around an MDA-MB-231 malignant epithelial breast cancer cell migrating through a type I collagen ECM. Both the normal and shear components of the resulting 3D stress field are in agreement with molecular and cellular scale processes involved in cell motility. Future application of our method, including detecting cancer cell malignancy and understanding the mechanics of each step of a cell's journey during cancer metastasis are discussed.