  We compute dynamic spin susceptibilities in the two-dimensional Hubbard model
using the method of Dual Fermions and provide comparison to lattice Monte Carlo
and cluster dynamical mean field theory. We examine the energy dispersion
identified by peaks in ${\rm Im}\chi(\omega,q)$ which define spin modes and
compare the exchange scale and magnon dispersion to neutron experiments on the
parent La$_2$CuO$_4$ cuprate. We present the evolution of the spin excitations
as a function of Hubbard interaction strengths and doping and explore the
particle-hole asymmetry of the spin excitations. We also study the correlation
lengths and the spin excitation dispersion peak structure and find a `Y'-shaped
dispersion similar to neutron results on doped HgBa$_2$CuO$_{4+\delta}$.
