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 ${𝖯}_{\omega }^{2n}(0,0)$. We prove that ${𝖯}_{\omega }^{2n}(0,0)$ is bounded by a random constant timesn−d/2 in d=2, 3, while it...