I don't know how useful it is in front-line research, but one of my former probability lecturers once made the following suggestion: "When in doubt, interchange the order of integration and summation. Miracles can occur."
On a more temperate note, although Fubini's theorem isn't a heuristic, when correctly applied, it's often worth ignoring integrability/convergence/a.e. issues and doing some formal manipulations, as long as one then goes through it again properly.
