Another approach to the theorem that could probably be rewritten to work in the PL category is the approach of Kirby and Melvin in Appendix C of the following paper:

MR1117149 (92e:57011) 
Kirby, Robion(1-CA); Melvin, Paul(1-BRYN)
The 3-manifold invariants of Witten and Reshetikhin-Turaev for sl(2,C). 
Invent. Math. 105 (1991), no. 3, 473–545. 

See Corollary C6.

The idea of this approach is as follows.  There is a famous $\mathbb{Z}/2$-invariant of homology $3$-spheres called the Rokhlin invariant.  The usual definition of this invariant is as follows.  Letting $M^3$ be a homology $3$-sphere, there exists a compact spin $4$-manifold $W^4$ with $\partial W^4 = M^3$.  Let $\sigma$ be the signature of $W^4$.  Rokhlin's theorem implies that modulo $16$, the value of $\sigma$ is independent of $W^4$.  Since $\sigma$ is divisible by $8$ for number-theoretic reasons (namely, van der Blij's lemma about quadratic forms), the value of $\sigma/8$ is well-defined modulo $2$.  

Using the Kirby calculus, Kirby and Melvin give a $3$-dimensional'' construction of the Rokhlin invariant, avoiding all mention of $4$-manifolds.  They then go backwards and use this to prove Rokhlin's theorem about $4$-manifolds.

Looking at their proof, Kirby and Melvin use smoothness in two ways.  The first is to prove that the Rokhlin invariant is well-defined.  But this is harmless since (by work of Moise) all PL $3$-manifolds can be smoothed in a unique way.  The second use of smoothness is to obtain a handlebody decomposition of the $4$-manifold.  But this should be easier in the PL category!

Existing proofs of Rokhlin's theorem for PL manifolds
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