We develop a new hybrid WKB technique to compute boundary-to-boundary scalar Green functions in asymptotically-AdS backgrounds in which the scalar wave equation is separable and is explicitly solvable in the asymptotic region. We apply this technique to a family of six-dimensional 1 8 -BPS asymptotically AdS 3 × S 3 horizonless geometries that have the same charges and angular momenta as a D1-D5-P black hole with a large horizon area. At large and intermediate distances, these geometries very closely approximate the extremal-BTZ × S 3 geometry of the black hole, but instead of having an event horizon, these geometries have a smooth highly-redshifted global-AdS 3 × S 3 cap in the IR. We show that the response function of a scalar probe, in momentum space, is essentially given by the pole structure of the highly-redshifted global-AdS 3 modulated by the BTZ response function. In position space, this translates into a sharp exponential black-hole-like decay for times shorter than N 1 N 5 , followed by the emergence of evenly spaced "echoes from the cap," with period ∼ N 1 N 5 . Our result shows that horizonless microstate geometries can have the same thermal decay as black holes without the associated information loss. arXiv:1905.05194v1 [hep-th]