The development of a reliable clinical technique for quantitative measurements of the parameters defining the BOLD effect, i.e., oxygen extraction fraction (OEF), and deoxygenated cerebral blood volume, dCBV, is needed to study brain function in health and disease. Herein we propose such a technique that is based on a widely available gradient recalled echo (GRE) MRI. Our method is based on GRE with multiple echoes and a model of signal decay (Yablonskiy, MRM 1998) that takes into account microscopic cellular (R2), mesoscopic (BOLD), and macroscopic (background field gradients) contributions to the GRE signal decay with additional accounting for physiologic fluctuations. Using 3 Tesla MRI, we generate high resolution quantitative maps of R2*, R2, R2', and tissue concentration of deoxyhemoglobin, the latter providing a quantitative version of SWI. Our results for OEF and dCBV in gray matter are in a reasonable agreement with the literature data. The proposed approach allows generating high resolution maps of hemodynamic parameters using clinical MRI. The technique can be applied to study such tissues as gray matter, tumors, etc.; however, it requires further development for use in tissues where extra- and intracellular compartments possess substantially different frequencies and relaxation properties (e.g., white matter).