  The generalised spectral dimension $D_{ S}(T)$ provides a powerful tool for
comparing different approaches to quantum gravity. In this work, we apply this
formalism to the classical spectral actions obtained within the framework of
almost-commutative geometry. Analysing the propagation of spin-0, spin-1 and
spin-2 fields, we show that a non-trivial spectral dimension arises already at
the classical level. The effective field theory interpretation of the spectral
action yields plateau-structures interpolating between a fixed spin-independent
$D_{ S}(T) = d_S$ for short and $D_{ S}(T) = 4$ for long diffusion times $T$.
Going beyond effective field theory
  the spectral dimension is completely dominated by the high-momentum
properties of the spectral action, yielding $D_{ S}(T)=0$ for all spins. Our
results support earlier claims that high-energy bosons do not propagate.
