Link is coloring a triforce, which consists of four equilateral triangles and is depicted below. He has three colors to use: gold, black, and green. So that it remains recognizable, he doesn't want to color any two triangles the same color if they share a side. How many different ways can he color the triforce? (Two colorings that differ by rotation are considered distinct.) [asy]
size(50);
draw((0,0)--(2,0)--(1,1.732)--cycle);
draw((1,0)--(1.5,.866)--(.5,.866)--cycle);
[/asy]
There are 3 ways to choose a color for the center.  Once the center color is chosen, for each of the remaining triangles there are 2 ways to choose a color (any color except the center color).  Therefore, there are $3\times 2\times 2\times 2 = \boxed{24}$ ways to color the triforce.