  We report on a study of QCD thermodynamics with three flavors of quarks,
using a Symanzik improved gauge action and the Asqtad O(a^2) improved staggered
quark action. Simulations were carried out with lattice spacings 1/4T, 1/6T and
1/8T both for three degenerate quarks with masses less than or equal to the
strange quark mass, m_s, and for degenerate up and down quarks with masses in
the range 0.1 m_s \leq m_{u,d} \leq 0.6 m_s, and the strange quark mass fixed
near its physical value. We present results for standard thermodynamics
quantities, such as the Polyakov loop, the chiral order parameter and its
susceptibility. For the quark masses studied to date we find a rapid crossover
rather than a bona fide phase transition. We have carried out the first
calculations of quark number susceptibilities with three flavors of sea quarks.
These quantities are of physical interest because they are related to
event-by-event fluctuations in heavy ion collision experiments. Comparison of
susceptibilities at different lattice spacings show that our results are close
to the continuum values.
