A Bott's theorem states that if a manifold admits a metric with all geodesics closed, then its homologies are isomorphic to homologies of one of the manifolds from the list: $S^n, \mathbb{RP}^n, \mathbb{CP}^n, \mathbb{HP}^n$ or $\mathbb{C}a\mathbb{P}^2$.

The problem of constructing such metrics is known to be extremely hard.

 So my particular interest is: what else is known about this problem in the case of dimension $3$?

For example, are all homological spheres known to admit such metrics? If a homological sphere (or a homological $\mathbb{RP}^3$ --- btw are there any examples, which are not $\mathbb{RP}^3$?)  has finite $\pi_1$, its universal covering is $S^3$, and I believe that all geodesics  will be closed in the metric induced by the standard metric on $S^3$. 

But I've just learned that there are examples of homological spheres with infinite fundamental group.  Their Thurston geometry is modeled by the universal covering of $\operatorname{SL}(2, \mathbb{R})$. It would be very interesting to know if such homological spheres admit metrics with all geodesics closed.