jmc

Expanding, we get $r^2+2+\frac{1}{r^2}=3,$ so $$r^2-1+\frac{1}{r^2}=0.$$Then $$r^3+\frac{1}{r^3}=\left(r+\frac{1}{r}\right)\left(r^2-1+\frac{1}{r^2}\right)=\boxed{0}.$$