  We present the first three-flavor lattice QCD calculations for $D\to \pi
l\nu$ and $D\to K l\nu$ semileptonic decays. Simulations are carried out using
ensembles of unquenched gauge fields generated by the MILC collaboration. With
an improved staggered action for light quarks, we are able to simulate at light
quark masses down to 1/8 of the strange mass. Consequently, the systematic
error from the chiral extrapolation is much smaller than in previous
calculations with Wilson-type light quarks. Our results for the form factors at
$q^2=0$ are $f_+^{D\to\pi}(0)=0.64(3)(6)$ and $f_+^{D\to K}(0) = 0.73(3)(7)$,
where the first error is statistical and the second is systematic, added in
quadrature. Combining our results with experimental branching ratios, we obtain
the CKM matrix elements $|V_{cd}|=0.239(10)(24)(20)$ and
$|V_{cs}|=0.969(39)(94)(24)$, where the last errors are from experimental
uncertainties.
