Bimodules in geometry
In "commutative geometry," I think bimodules tend to be a little concealed. People are more likely to talk about "correspondences" which are the space version of bimodules: A correspondence between spaces X and Y is a space Z with maps to X and Y.
When you think in this langauge, there are lots of examples you're missing. For example, the right notion of a morphism between two symplectic manifolds is a Lagrangian subvariety of their product, or even a manifold mapping to their product with Lagrangian image (maybe not embedded). See, for example, Wehrheim and Woodward's functoriality for Lagrangian correspondences in Floer homology
Similarly, correspondences are incredibly important in geometric representation theory. See, for example, the work of Nakajima on quiver varieties.
The theory of stacks also is at least partially founded on taking correspondences seriously as objects, and in particular being able to quotients by any (flat) correspondence.
This same philosophy also underlies groupoidification as studied by the Baez school (they tend to use the word "span" instead of "correspondence" but it's the same thing).
The Fourier-Mukai transform comes from a bimodule: the Poincaré bundle. Let $A$ be an abelian variety, the Poincaré bundle $\mathcal{P}$ is a vector bundle on $A \times \hat{A}$ coming from the fact that the points in the dual abelian variety $\hat{A}$ parametrize line bundles on $A$ ($\mathcal{P}$ is the universal family). In the Fourier-Mukai construction, $\mathcal{P}$ is used as a $\mathcal{O}_A$-$\mathcal{O}_{\hat{A}}$-bimodule to produce a functor between the derived categories of coherent sheaves on $A$ and $\hat{A}$ via a push-pull construction.
Here's a theorem from derived algebraic geometry: if A and B are A∞ algebras (think associative algebras) then giving an A-B-bimodule is the same as giving a functor from {right A-modules} to {right B-modules} which preserves colimits (equivalently, has a right adjoint). The correspondence sends AMB to the functor – ⊗A AMB. Under this correspondence, tensor product of bimodules over the middle algebra is realized by composition of functors.