I hope  that my ``answer'' will not be understood solely as a propaganda of my survey 
http://arxiv.org/abs/1101.2069 where I discussed 

(1) how, given geodesics, to reconstruct a connection (in both cases: when geodesics are parameterised and not parameterised)

(2) how, given a class od  projectively related  connections,  to reconstruct a metric,  and what are advantages of additional curvature assumptions on the metric 

(3) what is the freedom in reconstructing the metric by geodesics -- in particular I proved the statement wellknown to experts and mentionen by Robert Bryant in his comment that for generic metric the geodesics, even unparameterized, determine the metric.