  We study magnetic and charge susceptibilities in the half-filled
two-dimensional triangular Hubbard model within the dual fermion approximation
in the metallic, Mott insulating, and crossover regions of parameter space. In
the \textcolor{black}{insulating state}, we find strong spin fluctuations at
the K point at low energy corresponding to the \textcolor{black}{120$^{\circ}$}
antiferromagnetic order. These spin fluctuations persist into the metallic
phase and move to higher energy. We also present data for simulated neutron
spectroscopy and \textcolor{black}{spin-lattice} relaxation times, and perform
direct comparisons to inelastic neutron spectroscopy experiments on the
triangular material Ba$_8$CoNb$_6$O$_{24}$ and to the relaxation times on
$\kappa$-(ET)$_2$Cu$_2$(CN)$_3$. Finally, we present charge susceptibilities in
different areas of parameter space, which should correspond to
momentum-resolved electron-loss spectroscopy measurements on triangular
compounds.
