  We theoretically study the anomalous Hall effect (AHE) in perovskites with
antiferromagnetic (AFM) orderings. By studying the multiorbital Hubbard model
for $d$ electrons in perovskite transition metal oxides under the
GdFeO$_3$-type distortion within the Hartree-Fock approximation, we investigate
the behavior of intrinsic AHE owing to the atomic spin-orbit coupling via the
linear response theory. We consider the cases where there exist two ($d^2$) and
three ($d^3$) electrons in the $t_{2g}$ orbitals, and show that AFM ordered
states can exhibit AHE. In the $d^2$ case, $C$-type AFM states give rise to dc
AHE in metals and optical (finite-$\omega$) AHE in insulators accompanying
orbital ordering, while in the $d^3$ case, a $G$-type AFM insulating state
supports the optical AHE. By resolving the components in the spin patterns
compatible with the space group symmetry, we specify the collinear AFM
component to be responsible for the AHE, rather than the small ferromagnetic
component. We discuss the microscopic origin of the AHE: the collinear AFM spin
structure produces a nonzero Berry phase from the triangular units of the
lattice, activated by the complex orbital mixing terms owing to the
GdFeO$_3$-type distortion, and results in the microscopic Lorentz force.
