Here is the example I always give to the students. Take a random variable $X$ with $EX=0$  and $EX^3=0$, e.g. normal random variable with zero mean.  Take $Y=X^2$. It is clear that $X$ and $Y$ are related, but 

$$cov(X,Y)=EXY-EX\cdot EY=EX^3=0.$$