We present the results of three-dimensional femtoscopic analyses for charged and neutral kaons recorded by ALICE in Pb-Pb collisions at sNN=2.76 TeV. Femtoscopy is used to measure the space-time characteristics of particle production from the effects of quantum statistics and final-state interactions in two-particle correlations. Kaon femtoscopy is an important supplement to that of pions because it allows one to distinguish between different model scenarios working equally well for pions. In particular, we compare the measured three-dimensional kaon radii with a purely hydrodynamical calculation and a model where the hydrodynamic phase is followed by a hadronic rescattering stage. The former predicts an approximate transverse mass (mT) scaling of source radii obtained from pion and kaon correlations. This mT scaling appears to be broken in our data, which indicates the importance of the hadronic rescattering phase at LHC energies. A kT scaling of pion and kaon source radii is observed instead. The time of maximal emission of the system is estimated by using the three-dimensional femtoscopic analysis for kaons. The measured emission time is larger than that of pions. Our observation is well supported by the hydrokinetic model predictions.