Geodesics in a point mass universe This question may reflect my (lack of) knowledge about general relativity, please ask for any clarifications or note any corrections in the comments and I'll try to address them. 
The Schwarzschild solution to the Einstein's field equations gives the solution to a a non-rotating, spherically symmetric uncharged mass distribution in a static vacuum. From this solution, we can compute geodesics from one point to another.
Is it possible (analytically or numerically) to compute the geodesics from one point to another under the influence of a set of static point masses? The point masses can assume arbitrary geometry, but like the Schwarzschild problem they are uncharged and non-rotating. 
 A: There are no static solutions for multiple uncharged point masses. This fact becomes apparent, if we note that the point mass initially at rest will start moving in the gravitational field of the rest of masses. 
So if one is looking for a static spacetime with several point masses, then one have to provide additional force to hold the masses stationary and counteract the gravitational attraction. 
One such possibility are the Israel-Khan solutions, which describe finitely many collinear Schwarzschild black holes in static equilibrium. The forces holding them apart arise from conical deficits in the form of 'struts' between the black holes, or cosmic strings extending to infinity. 
Another mechanism to counteract gravitational attraction is electrostatic repulsion. This leads us to Majumdar-Papapetrou (MP) solutions describing the set of charged extremal (which means that charge is equal to mass in natural units) black holes.
The geodesics in this class of metric have been studied (along with trajectories of charged particles). 
Here is the paper describing the chaotic dynamics of a test particle moving in such a spacetime:

Cornish, N.J., Dettmann, C.P. &  Frankel, N.E.  Fractal basins and chaotic trajectories in multi-black hole space-times. Phys. Rev. D50:618-621 (1994), DOI:10.1103/PhysRevD.50.R618, arXiv:gr-qc/9402027.

Besides providing the equations of motion for the test particle (it can also be charged) in the MP metric (which are then integrated numerically), this paper describes the previous work done in the field:

Chandrasekhar [12] and Contopoulos [13], [14] have investigated the timelike and null
  geodesics of the two black hole system from the point of view of the periodic orbits and
  the weak field limit. For particles with elliptic (bound) energies the trajectories fall into several categories. There are stable periodic and quasiperiodic orbits, chaotic orbits trapped between periodic orbits, trajectories which fall into one or other of the black holes, and chaotic trajectories which lie on the boundary of these regions. This structure, together with the fact that the weak field limit is integrable, makes the MP geodesics problem a particularly interesting example of chaos in General Relativity.

So, if you are interested in analytical results concerning these metrics you could turn to this referenced works.
