  Using density functional theory calculations, we demonstrate that the effect
of anisotropic magnetoresistance (AMR) can be enhanced by orders of magnitude
with respect to conventional bulk ferromagnets in junctions containing
molecules sandwiched between ferromagnetic leads. We study ballistic transport
in metal-benzene complexes contacted by $3d$ transition-metal wires. We show
that the gigantic AMR can arise from spin-orbit coupling effects in the leads,
drastically enhanced by orbital-symmetry filtering properties of the molecules.
We further discuss how this molecular anisotropic magnetoresistance (MAMR) can
be tuned by proper choice of materials and their electronic properties.
