How many different three-step paths along the edges of a cube are there that take you from vertex $A$ to vertex $B$? (A step is from a vertex to an adjacent vertex sharing an edge.)

[asy]
size(101);
import three;
currentprojection=orthographic(1/2,-1,1/4);
draw((0,0,0)--(1,0,0)--(1,0,1)--(0,0,1)--(0,0,0));
draw((1,0,0)--(1,1,0)--(1,1,1)--(1,0,1));
draw((1,1,1)--(0,1,1)--(0,0,1));
label("A",(0,0,1),(-1,0,0)); label("B",(1,1,0),(1,0,0));

[/asy]
There are $3$ points we can get to from $A$.  Each of these connects to two points adjacent to $B$ and to $A$.  We can't go back to $A$ and then get to $B$ in one step, but we can choose either of the other two points.  So there are $3(2)=\boxed{6}$ paths from $A$ to $B$.