Yes, there is a geometric approach, it is called, as one might guess, geometric quantization. If you read a literature on the subject, you'll find a lot of beautiful results, I'll describe the basic idea below.
The geometric quantization works with a phase space rather then configuration, i.e. you start with a symplectic manifold. It's actually a more general case because once you have the configuration manifold, the corresponding symplectic manifold is just its cotangent bundle with natural symplectic structure.
You should fix extra geometric data -- a polarization. It's a half-rank integrable distribution, so locally on the phase space you have a foliation, which defines what direction corresponds to "coordinates" and what -- to momenta. For example in the usual "p,q" case you take half-rank distribution, i.e. half-rank subbundle of tangent bundle, generated by ${\partial\over\partial p}$, so it generates "momentum direction" and the transverse direction corresponds to "coordinates".
Then you build a prequantization line bundle, which is a hermitean line bundle with hermitean connection, such that its curvature is proportional to the symplectic form (the usual normalization ${i\over 2\pi}R = \omega$, where $R$ -- curvature, $\omega$ -- symplectic structure), so that symplectic form is just the first Chern class of this line bundle. You do not need polarization for this step. But you actually need the first Chern class of the bundle to be integer, which can be considered as a general case (which corresponds in the compact case to the condition that the phase space has integer volume in the units of $(2\pi\hbar)^{dim/2}$).
Then you consider the space of sections of this bundle and take it's subspace, annihilated but the connection (covariant derivative), taken along the polarization. What obtained (or, more precisely, $L^2$-closure of square-integrable sections from the obtained subspace) is defined to be the Hilbert space of the problem. In the standard example, you take functions on the phase space, annihilated by ${\partial\over\partial p}$, which are just functions $\psi(q)$ depending on coordinates only, and then consider square-integrable functions of coordinates.
Next for each (smooth) function on the manifold you build operators on that space. There are different existing prescriptions for that, I'll not cover it here.
Should also note that there exists another version of geometric quantization -- Berezin-Toeplitz quantization -- where you don't need polarization in a sense described above. Instead you define the Hilbert space either as as a kernel of the $Spin^c$ Dirac operator, or as a kernel of the "renormalized Bochner-Laplasian" operator. This approach is pretty computable, especially in the case when the symplectic manifold is actually Kahler (in such case standard geometric quantization works well too, since there is a canonical holomorphic polarization).
Also have to add, that since there is no global notion of coordinate in general case, there is no "coordinate operator", instead you build operators, corresponding to smooth functions on the manifold.
And one more comment -- it is tentative to say, that path integral could provide such a coordinate-free description. But in fact path integral is not a well-defined notion by itself, there's no well-defined and invariant notion of integration over the infinite-dimensional space of paths in general. To define path integral you need to fix some additional data, you need to "regularize" it (discretize, mode-expand, whatever). In many cases that will include fixing coordinates as well. Also, in general case this regularization can break coordinate invariance of the phase space.
Path integral can't be considered as an honest prescription. In general it's just a heuristic instrument used by physicists in cases, when they have no better tools to use. Though in some particular examples path integral can be justified.
