  We present a rotationally invariant Hirsch-Fye quantum Monte Carlo algorithm
in which the spin rotational invariance of Hund's exchange is approximated by
averaging over all possible directions of the spin quantization axis. We employ
this technique to perform benchmark calculations for the two- and three-band
Hubbard models on the infinite-dimensional Bethe lattice. Our results agree
quantitatively well with those obtained using the continuous-time quantum Monte
Carlo method with rotationally invariant Coulomb interaction. The proposed
approach is employed to compute the electronic and magnetic properties of
paramagnetic $\alpha$ iron and nickel. The obtained Curie temperatures agree
well with experiment. Our results indicate that the magnetic transition
temperature is significantly overestimated by using the density-density type of
Coulomb interaction.
