The PHENIX experiment at the BNL Relativistic Heavy Ion Collider (RHIC) has measured J/psi production for rapidities -2.2 < y < 2.2 in Au+Au collisions at root s(NN)=200 GeV. The J/psi invariant yield and nuclear modification factor R-AA as a function of centrality, transverse momentum, and rapidity are reported. A suppression of J/psi relative to binary collision scaling of proton-proton reaction yields is observed. Models which describe the lower energy J/psi data at the CERN Super Proton Synchrotron invoking only J/psi destruction based on the local medium density predict a significantly larger suppression at RHIC and more suppression at midrapidity than at forward rapidity. Both trends are contradicted by our data.