Two-dimensional(2D) and three-dimensional (3D) hydrodynamical simulations have been intensively performed recently to clarify the accretion-flow structure in the low-radiation-efficiency limit. However, the results depend critically on the parameterized magnitude of the viscosity, which, in principle, should be determined self-consistently by MHD simulations. We analyzed the structure of 3D MHD accretion flows initially threaded by weak toroidal magnetic fields, and found for the first time large-scale convective motions dominating near to the black hole. Radial profiles of each physical quantity include: the density, rho proportional to r(-0.5); radial velocity, v(r) proportional to r-(1.5); temperature, T proportional to r(-1.0); and field strength, B-2 proportional to r(-1.5). Although the flow structure, itself, appears to be similar to those obtained by hydrodynamic simulations, the observational appearance is distinct. Unlike non-magnetic models, in which radiation is dominant at the outermost convective zones because of outward energy flow by convection, substantial accretion energy can be released in the vicinity of a black hole in MHD flow via magnetic reconnection. Such reconnection leads to sporadic flare events, thus producing variability in out-going radiation, as is commonly observed in objects with black-hole accretion.