  We study the normal state and the superconducting transition in the
Attractive Hubbard Model in three dimensions, using self-consistent
diagrammatics. Our results for the self-consistent $T$-matrix approximation are
consistent with 3D-XY power-law critical scaling and finite-size scaling. This
is in contrast to the exponential 2D-XY scaling the method was able to capture
in our previous 2D calculation. We find the 3D transition temperature at
quarter-filling and $U=-4t$ to be $T_c=0.207t$. The 3D critical regime is much
narrower than in 2D and the ratio of the mean-field transition to $T_c$ is
about 5 times smaller than in 2D. We also find that, for the parameters we
consider, the pseudogap regime in 3D (as in 2D) coincides with the critical
scaling regime.
