  Local topological markers, topological invariants evaluated by local
expectation values, are valuable for characterizing topological phases in
materials lacking translation invariance. The Chern marker -- the Chern number
expressed in terms of the Fourier transformed Chern character -- is an easily
applicable local marker in even dimensions, but there are no analogous
expressions for odd dimensions. We provide general analytic expressions for
local markers for free-fermion topological states in odd dimensions protected
by local symmetries: a Chiral marker, a local $\mathbb Z$ marker which in case
of translation invariance is equivalent to the chiral winding number, and a
Chern-Simons marker, a local $\mathbb Z_2$ marker characterizing all nonchiral
phases in odd dimensions. We achieve this by introducing a one-parameter family
$P_{\vartheta}$ of single-particle density matrices interpolating between a
trivial state and the state of interest. By interpreting the parameter
$\vartheta$ as an additional dimension, we calculate the Chern marker for the
family $P_{\vartheta}$. We demonstrate the practical use of these markers by
characterizing the topological phases of two amorphous Hamiltonians in three
dimensions: a topological superconductor ($\mathbb Z$ classification) and a
topological insulator ($\mathbb Z_2$ classification).
