I would use ParametricRegion. In line of principle, I would expect this to give a hemisphere:
DiscretizeRegion@ParametricRegion[
{FromSphericalCoordinates[{\[Rho], \[Theta], \[Phi]}],
0 <= \[Rho] <= 3 && 0 <= \[Theta] <= 2 \[Pi] &&
0 <= \[Phi] <= Pi/2},
{\[Rho], \[Theta], \[Phi]}]
Instead, I get a cube. The two dimensional analogue works fine:
DiscretizeRegion@
ParametricRegion[{FromPolarCoordinates[{\[Rho], \[Theta]}],
0 <= \[Rho] <= 3 && 0 <= \[Theta] <= 2 \[Pi]},
{\[Rho], \[Theta]}]
