A nice, direct combinatorial construction was given by Gaifullin, see his papers on the arXiv (equivalently: Explicit construction of manifolds realizing the prescribed homology classes, Realisation of cycles by aspherical manifolds and Configuration spaces, bistellar moves, and combinatorial formulae for the first Pontryagin class). A drawback of this approach is that if you think of it as realizing some multiple $m\alpha$ of the given integral homology class $\alpha$ by an oriented smooth manifold, then $m$ is not bounded in terms of the dimension of $\alpha$.
There has also been another geometric approach. Thom also proved that $\bmod2$ homology classes are representable by maps of smooth (possibly unorientable) manifolds. This was reproved geometrically in
S. Buoncristiano and D. Hacon, An elementary geometric proof of two theorems of Thom, Topology 20 (1981), no. 1, 97–99 (Core pdf)
The other theorem of their title is that unoriented bordism is determined by Stiefel-Whitney numbers, and it is used in their proof that mod 2 homology classes are representable by smooth manifolds.
I believe the same geometric argument should also work to show that rational homology classes are representable by oriented smooth manifolds - modulo the fact that Pontryagin numbers determine oriented bordism tensored by $\Bbb Q$. This fact I'm afraid I don't know how to prove geometrically (for some proof, see e.g. the Milnor-Stasheff book). But note that in a subsequent paper Buouncristiano and Hacon also gave a geometric proof that Chern numbers determine complex bordism (Ann. of Math., 118 (1983), 1-7 https://doi.org/10.2307/2006950). Their other papers may also be of interest if you care about geometric proofs of classical results on bordism.
