  We study bond percolation on the simple cubic (SC) lattice with various
combinations of first, second, third, and fourth nearest-neighbors by Monte
Carlo simulation. Using a single-cluster growth algorithm, we find precise
values of the bond thresholds. Correlations between percolation thresholds and
lattice properties are discussed, and our results show that the percolation
thresholds of these and other three-dimensional lattices decrease monotonically
with the coordination number $z$ quite accurately according to a power law
$p_{c} \sim z^{-a}$, with exponent $a = 1.111$. However, for large $z$, the
threshold must approach the Bethe lattice result $p_c = 1/(z-1)$. Fitting our
data and data for lattices with additional nearest neighbors, we find
$p_c(z-1)=1+1.224 z^{-1/2}$.
