We consider the nearest-neighbor simple random walk on ℤd, d≥2, driven by a field of bounded random conductances ωxy∈[0, 1]. The conductance law is i.i.d. subject to the condition that the probability of ωxy>0 exceeds the threshold for bond percolation on ℤd. 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 $\mathsf{P}_{\omega}^{2n}(0,0)$ . We prove that $\mathsf{P}_{\omega}^{2n}(0,0)$ is bounded by a random constant times n−d/2 in d=2, 3, while it is o(n−2) in d≥5 and O(n−2log n) in d=4. By producing examples with anomalous heat-kernel decay approaching 1/n2, we prove that the o(n−2) bound in d≥5 is the best possible. We also construct natural n-dependent environments that exhibit the extra log n factor in d=4.