Tunneling anisotropic magnetoresistance (TAMR) effect, discovered recently in (Ga,Mn)As ferromagnetic semiconductors, arises from spin-orbit coupling and reflects the dependence of the tunneling density of states in a ferromagnetic layer on orientation of the magnetic moment. Based on ab initio relativistic calculations of the anisotropy in the density of states we predict sizable TAMR effects in room-temperature metallic ferromagnets. This opens prospect for new spintronic devices with a simpler geometry as these do not require antiferromagnetically coupled contacts on either side of the tunnel junction. We focus on several model systems ranging from simple hcp-Co to more complex ferromagnetic structures with enhanced spin-orbit coupling, namely bulk and thin film L1$_0$-CoPt ordered alloys and a monatomic-Co chain at a Pt surface step edge. Reliability of the predicted density of states anisotropies is confirmed by comparing quantitatively our ab initio results for the magnetocrystalline anisotropies in these systems with experimental data.