Yes, all hyperbolic three-manifold fundamental groups can be generated by loxodromic elements. For, suppose that $\Gamma = \{ \gamma_i \}$ is a generating set. Take $\gamma$, a loxodromic element which is "sufficiently long" compared to the elements $\Gamma$. Then the set $\{ \gamma \} \cup \{\gamma \cdot \gamma_i\}$ generates and consists only of loxodromic elements.
A more delicate argument will give loxodromic generating sets, of size two, of (say) the twist knot groups.
