Look at this decomposition of a scalar field in 3D space:
$\phi(r) = \oint_{\partial V} G(r-r') \cdot \hat n |dS'| \phi(r') + \int_V G(r-r') \; |dV'| \; \nabla' \phi(r')$
where $G(r) = (r-r')/4\pi|r-r'|^3$ is a Green's function. Note that when $\nabla \phi = 0$ on the volume, the value of the function at a point is determined entirely by the values of the function on the boundary.
Sound familiar? It should. The residue theorem is just a special case of this basic concept. Complex analysis gets it backwards compared to vector calculus, where to be determined completely by boundary values, a field must have zero gradient (or both zero curl and divergence) rather than being complex differentiable, but the concepts are the same. The analogues of complex analytic functions are those whose vector derivatives (whether they be gradients, divergences, or curls--or their analogues in higher dimensions) are zero, so that the volume term on the RHS vanishes, and the function is entirely determined by its boundary values.
You asked about the Cauchy-Riemann condition. It can be shown that the generalized condition is one that ensures vector fields have no divergence or curl.
In short, the residue theorem does have generalizations to real vector spaces, just in guises that may be harder to recognize, as vector calculus is actually quite a bit more general--in terms of the kinds of functions it considers--than complex analysis. Vector fields with arbitrary sources are common, while holomorphic functions are essentially ones with no divergence or curl, and meromorphic ones are analogous to vector fields generated by only point sources.