Francesco's answer is completely general.  As he emphasized, we should do everything in an intrinsic way so that there is a canonical choice of Theta.  I hope I get this straight since it is very pretty.

Indeed this is Riemann's classical proof of Jacobi inversion.  If L is any degree g line bundle, then K-L has degree g-2, so the translate C+(K-L) lies in Pic^(g-1). Then the pullback of O(Theta) = O(W(g-1)) by this translation map C-->Pic^(g-1)(C) is L.  (This also shows that this map induces an isomorphism from line bundles on Pic^(g-1)(C) of chern class [Theta], with line bundles on C of degree g.)

I.e. consider this as the problem of intersecting the curve translate C+(K-L) with the divisor W(g-1).  We know L = O(p1+...+pg) for some points pi, and that the intersection has degree g.  So if the intersection is proper, there exist g points p such that (p+ K -p1-...-pg) lies on W(g-1), i.e. such that

p + K-p1-...-pg = q1+...+qg-1, for some q's.

But this is true for p equal to any pi, i=1,....,g, if for p = pi0 we take 
the q's to be such that O(q1+...+qg-1) = O(K- sum of pi for i ≠ i0).  (Such q's exist by Riemann-Roch.)


Thus intersecting C+(K-L) with W(g-1) was Riemann's way of finding a divisor D such that O(D) = L.  Your question is the same, but phrased in terms of line bundles.  The common practice in many books of always working in Pic^0(C) via translating by a fixed point may obscure the intrinsic nature of the statements.  In fact I seem to be guilty of this in my argument on pp.386-7 of Lectures on Riemann Surfaces, (World Scientific Publishers 1989, ed. by Cornalba, Gomez-Mont and Verjovsky), where I take the strange approach of deducing this intrinsic version from a clunky non intrinsic version, thus making it look less comprehensible.