Is it true that an abelian group is cyclic iff every $n$-torsion subgroup is cyclic? Obviously if it is cyclic every subgroup, including the torsion subgroups, is cyclic. I'm having trouble figuring if the converse is true.

No.  Consider the circle group $\{e^{i\theta}\mid\theta\in [0,2\pi)\}$.
For the finite case, it's true.  For, if $G$ is not cyclic,  it contains a subgroup isomorphic to $\Bbb Z_p×\Bbb Z_p$.  But then, the $p$-torsion subgroup is not cyclic.  
