  We present preliminary results from exploring the phase diagram of finite
temperature QCD with three degenerate flavors and with two light flavors and
the mass of the third held approximately at the strange quark mass. We use an
order $\alpha_s^2 a^2, a^4$ Symanzik improved gauge action and an order
$\alpha_s a^2, a^4$ improved staggered quark action. The improved staggered
action leads to a dispersion relation with diminished lattice artifacts, and
hence better thermodynamic properties. It decreases the flavor symmetry
breaking of staggered quarks substantially, and we estimate that at the
transition temperature for an $N_t=8$ to $N_t=10$ lattice {\em all} pions will
be lighter than the lightest kaon. Preliminary results on lattices with
$N_t=4$, 6 and 8 are presented.
