  We study the Hawking radiation emitted into the bulk by $(D+1)$-dimensional
Schwarzschild black holes. It is well-known that the black-hole spectrum
departs from exact blackbody form due to the frequency dependence of the
`greybody' factors. For intermediate values of $D$ ($3\leq D\lesssim10$), these
frequency-dependent factors may significantly modify the spectrum of the
emitted radiation. However, we point out that for $D\gg1$, the typical
wavelengths in the black-hole spectrum are much {\it shorter} than the size of
the black hole. In this regime, the greybody factors are well described by the
geometric-optics approximation according to which they are almost
frequency-independent. Following this observation, we argue that for
higher-dimensional black holes with $D\gg1$, the total power emitted into the
bulk should be well approximated by the analytical formula for perfect
blackbody radiation. We test the validity of this analytical prediction with
numerical computations.
