  We employ a recently developed variant of the functional renormalization
group method for spin systems, the so-called pseudo Majorana functional
renormalization group, to investigate three-dimensional spin-1/2 Heisenberg
models at finite temperatures. We study unfrustrated and frustrated Heisenberg
systems on the simple cubic and pyrochlore lattices. Comparing our results with
other quantum many-body techniques, we demonstrate a high quantitative accuracy
of our method. Particularly, for the unfrustrated simple cubic lattice
antiferromagnet ordering temperatures obtained from finite-size scaling of
one-loop data deviate from error controlled quantum Monte Carlo results by
$\sim5\%$ and we further confirm the established values for the critical
exponent $\nu$ and the anomalous dimension $\eta$. As the PMFRG yields results
in good agreement with QMC, but remains applicable when the system is
frustrated, we next treat the pyrochlore Heisenberg antiferromagnet as a
paradigmatic magnetically disordered system and find nearly perfect agreement
of our two-loop static homogeneous susceptibility with other methods. We
further investigate the broadening of pinch points in the spin structure factor
as a result of quantum and thermal fluctuations and confirm a finite width in
the extrapolated limit $T\rightarrow0$. While extensions towards higher loop
orders $\ell$ seem to systematically improve our approach for magnetically
disordered systems we also discuss subtleties when increasing $\ell$ in the
presence of magnetic order. Overall, the pseudo Majorana functional
renormalization group is established as a powerful many-body technique in
quantum magnetism with a wealth of possible future applications.
