A rational elliptic surface is the blow-up of P2 in the nine base points of a pencil of cubics. The pencil then lifts as a fibration of the surface by elliptic curves. I show that the complement of any fiber F admits families of complete Calabi-Yau metrics, whose asymptotic geometry depends in a delicate way on the monodromy of the fibration around F. If F is smooth, these metrics all converge to flat cylinders at an exponential rate, and in that case I give a complete description of the local Einstein deformation space.