The first reference attempts to solve this problem but only gives a partial answer. The second reference shows that every 3-manifold can be obtained by surgery on a link (but does not discuss Kirby calculus).
MR1075370 (91k:57019) Rêgo, Eduardo ; de Sá, Eugénia César . Special Heegaard diagrams and the Kirby calculus. Topology Appl. 37 (1990), no. 1, 11--24.
MR0809959 (87f:57016) Rourke, Colin . A new proof that $\Omega\sb 3$ \Omega_3$ is zero. J. London Math. Soc. (2) 31 (1985), no. 2, 373--376.
I apologise if you are already aware of these references.