  We present extensive new \emph{ab initio} path integral Monte Carlo (PIMC)
simulations of normal liquid $^3$He without any nodal constraints. This allows
us to study the effects of temperature on different structural properties like
the static structure factor $S(\mathbf{q})$, the momentum distribution
$n(\mathbf{q})$, and the static density response function $\chi(\mathbf{q})$,
and to unambiguously quantify the impact of Fermi statistics. In addition, the
dynamic structure factor $S(\mathbf{q},\omega)$ is rigorously reconstructed
from imaginary-time PIMC data, and we find the familiar phonon-maxon-roton
dispersion that is well known from $^4$He and has been reported previously for
two-dimensional $^3$He films [Nature \textbf{483}, 576-579 (2012)]. The
comparison of our new results for both $S(\mathbf{q})$ and
$S(\mathbf{q},\omega)$ to neutron scattering measurements reveals an excellent
agreement between theory and experiment.
