We present a search for the rare decays B(+) -> K(+) v (v) over bar and B(0) -> K(0) v (v) over bar using 459 x 10(6) B (B) over bar pairs collected with the BABAR detector at the SLAC National Accelerator Laboratory. Flavor-changing neutral-current decays such as these are forbidden at tree level but can occur through one-loop diagrams in the standard model (SM), with possible contributions from new physics at the same order. The presence of two neutrinos in the final state makes identification of signal events challenging, so reconstruction in the semileptonic decay channels B -> D((*)) lv of the B meson recoiling from the signal B is used to suppress backgrounds. We set an upper limit at the 90% confidence level (CL) of 1.3 x 10(-5) on the total branching fraction for B(+) -> K(+) v (v) over bar, and 5.6 x 10(-5) for B(0) -> K(0) v (v) over bar. We additionally report 90% CL upper limits on partial branching fractions in two ranges of dineutrino mass squared for B(+) -> K(+) v (v) over bar.