It is well known that, given a quantum field in Minkowski space, one can define Wigner functions ${f}_{W}^{N}$(${x}_{1}$,${p}_{1}$,...,${x}_{N}$,${p}_{N}$) which (a) are convenient to analyze since, unlike the field itself, they are c-number quantities and (b) can be interpreted in a limited sense as ``quantum distribution functions.'' Recently, Winter and Calzetta, Habib and Hu have shown one way in which these flat-space Wigner functions can be generalized to a curved-space setting, deriving thereby approximate kinetic equations which make sense ``quasilocally'' for ``short-wavelength modes.'' This paper suggests a completely orthogonal approach for defining curved-space Wigner functions which generalizes instead an object such as the Fourier-transformed ${f}_{W}^{1}$(k,p), which is effectively a two-point function viewed in terms of the ``natural'' creation and annihilation operators ${a}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$(p-(1/2k) and a(p+(1/2k). The approach suggested here lacks the precise phase-space interpretation implicit in the approach of Winter or Calzetta, Habib, and Hu, but it is useful in that (a) it is geared to handle any ``natural'' mode decomposition, so that (b) it can facilitate exact calculations at least in certain limits, such as for a source-free linear field in a static spacetime.