We consider the nearest-neighbor simple random walk on $\Z^d$, $d\ge2$,
driven by a field of bounded random conductances $\omega_{xy}\in[0,1]$.
The conductance law is i.i.d. subject to the condition that the probability
of $\omega_{xy}>0$ exceeds the threshold for bond percolation on $\Z^d$.
For environments in which the origin is connected to infinity by bonds
with positive conductances, we study the decay of the $2n$-step return
probability $P_\omega^{2n}(0,0)$. We prove that $P_\omega^{2n}(0,0)$ is
bounded by a random constant times $n^{-d/2}$ in $d=3D2,3$, while it is
$o(n^{-2})$ in $d\ge5$ and $O(n^{-2}\log n)$ in $d=3D4$. By producing
examples with anomalous heat-kernel decay approaching $1/n^2$ we prove
that the $o(n^{-2})$ bound in $d\ge5$ is the best possible. We also
construct natural $n$-dependent environments that exhibit the extra $\log
n$ factor in $d=3D4$.