Fluctuations of conserved quantities such as baryon number, charge, and strangeness are sensitive to the correlation length of the hot and dense matter created in relativistic heavy-ion collisions and can be used to search for the QCD critical point. We report the first measurements of the moments of net-kaon multiplicity distributions in Au+Au collisions atsqrt(sNN) = 7.7 , 11.5, 14.5, 19.6, 27, 39, 62.4, and 200 GeV. The collision centrality and energy dependence of the mean ( M ), variance ( σ2 ), skewness ( S ), and kurtosis ( κ ) for net-kaon multiplicity distributions as well as the ratio σ2/M and the products Sσ and κσ2 are presented. Comparisons are made with Poisson and negative binomial baseline calculations as well as with UrQMD, a transport model (UrQMD) that does not include effects from the QCD critical point. Within current uncertainties, the net-kaon cumulant ratios appear to be monotonic as a function of collision energy.