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.
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