We propose and develop the theoretical framework for a new experimental technique for constructing Poincaré maps in three-dimensional flows exhibiting chaotic advection. The technique is non-intrusive and, thus, simple to implement. Planar laser-induced fluorescence (LIF) is employed to collect a sufficiently long sequence of instantaneous light intensity fields on the plane of section of the Poincaré map (defined by the laser sheet). The chains of unmixed (regular) islands in the flow are visualized by time-averaging the instantaneous images and plotting iso-contours of the resulting mean light intensity field. A rigorous theoretical justification for this technique is derived using concepts from ergodic theory. We demonstrate the capabilities of the method by applying it to visualize the rich Lagrangian dynamics within steady vortex breakdown bubbles in a closed cylinder with a rotating bottom. The experimental results are shown to be in excellent agreement with numerical simulations.