What fraction of this square region is shaded?  Stripes are equal in width, and the figure is drawn to scale.
[asy] unitsize(8); fill((0,0)--(6,0)--(6,6)--(0,6)--cycle,black); fill((0,0)--(5,0)--(5,5)--(0,5)--cycle,white); fill((0,0)--(4,0)--(4,4)--(0,4)--cycle,black); fill((0,0)--(3,0)--(3,3)--(0,3)--cycle,white); fill((0,0)--(2,0)--(2,2)--(0,2)--cycle,black); fill((0,0)--(1,0)--(1,1)--(0,1)--cycle,white); draw((0,6)--(0,0)--(6,0)); [/asy]
$\text{(A)}\ \dfrac{5}{12} \qquad \text{(B)}\ \dfrac{1}{2} \qquad \text{(C)}\ \dfrac{7}{12} \qquad \text{(D)}\ \dfrac{2}{3} \qquad \text{(E)}\ \dfrac{5}{6}$
Fill in grid lines to make the shape look like a small 6x6 checkerboard.
There are $6\times6 = 36$ squares total.  The smallest shaded region has $3$ squares.  The medium shaded region has $7$ squares.  The largest shaded region has $11$ squares.  The total shaded area is $3 + 7 + 11 = 21$ squares.
Thus, $\frac{21}{36} = \frac{7}{12}$ of the square is shaded, and the answer is $\boxed{C}$