Claim 1: If $x\in H_2X$ is a homology class, then there exists a 2-dimensional CW complex $K$ and a map $f:K\to X$, such that $x$ is in the image of $f_*:H_2K\to H_2X$.
Claim 2: If $X$ is simply connected with $\pi_2X=0$, then every map $K\to X$ from a 2-dimensional CW complex is null homotopic.
Claim 1 can be proved by taking $K$ to be a finite union of triangles attached together along suitable edges, built by considering an explicit cocycle representing $x$. Claim 2 is easy.
Of course, this is just the $n=3$ case of the standard proof of "easy Hurewicz" (that $\pi_kX=0$ for $k< n$ implies $H_kX=0$ for
$k<n$; non-easy Hurewicz is the statement relating the non-trivial groups in dimension $n$, but you don't want that part.)