A nice, direct combinatorial construction was given by Gaifullin, see his [papers][1] [on the][2] [arXiv][3] (equivalently: _[Explicit construction of manifolds realizing the prescribed homology classes](https://arxiv.org/abs/0712.1709)_, _[Realisation of cycles by aspherical manifolds](https://doi.org/10.4213/rm9198)_ and _[
Configuration spaces, bistellar moves, and combinatorial formulae for the first Pontryagin class](https://doi.org/10.1134/S0081543810010074)_). 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](https://core.ac.uk/download/pdf/82811869.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.


  [1]: https://arxiv.org/abs/0712.1709
  [2]: https://arxiv.org/abs/0806.3580
  [3]: https://arxiv.org/abs/0912.3933