The tunneling method for stationary black holes in the Hamilton-Jacobi variant is reconsidered in the light of some critiques that have been moved against. It is shown that once the tunneling trajectories have been correctly identified the method is free from internal inconsistencies, it is manifestly covariant, it allows for the extension to spinning particles and it can even be used without solving the Hamilton-Jacobi equation. These conclusions borrow support on a simple analytic continuation of the classical action of a pointlike particle, made possible by the unique assumption that it should be analytic in the complexified Schwarzschild or Kerr-Newman space-time. A more general version of the Parikh-Wilczek method will also be proposed along these lines.