There is a "half-geometric" observation that since $$ \pi_3^s\cong H_3(\tilde A_N),\ N\gg0 $$ where $\tilde A_N$ is the universal central extension of the $N$th alternating group, elements of $\pi_3^s$ might be described by certain 3-cycles (hopefully of geometric origin).
Seemingly this geometry is easier to discern after the embedding $\tilde A_N\hookrightarrow\mathrm{St}_N(\mathbb Z)$ into the Steinberg group of the integers.
(For a ring $R$, the group $\mathrm{St}_N(R)$ is the universal central extension of $\mathrm E_N(R)$ - the group which for most decent rings is the same as $\mathrm{SL}_N(R)$; one has $K_3(R)\cong H_3(\mathrm{St}(R))$ which in good cases stabilizes after some $\mathrm{St}_N(R)$.)
In "The generalized Grassmann invariant" (late 70ies) K. Igusa introduced technique of pictures to construct a homomorphism $$ \chi:K_3(\mathbb Z[\pi])\to H_0(\pi;\mathbb F_2[\pi]); $$ in his own words, "The definition of $\chi$ comes from very intuitive geometric considerations, but unfortunately the algebraic analogue is rather clumsy."
Anyway geometry is still there as this map is strongly related to pseudoisotopy of compact manifolds (with $\pi_1=\pi$ and $\pi_2=0$). The image of $\pi_3^s(B\pi_+)\to K_3(\mathbb Z[\pi])$ is in the kernel of $\chi$, and for trivial $\pi$ all this enabled him to detect an element of order 48 in $K_3(\mathbb Z)$; $\pi_3^s$ is a subgroup of index 2 there. His picture representing a generator of $K_3(\mathbb Z)$ ![Igusa's picture for the generator of K_3(Z)][1]
has been haunting me for decades. It shows something like "$1/2$ of the generator of $\pi_3^s$ living outside $\tilde A_N\hookrightarrow\mathrm{St}_N(\mathbb Z)$" and there surely must be something simpler which represents the generator of $\pi_3^s$ inside $\tilde A_N$ itself. This should not be that difficult as $\tilde A_N$ comes equipped with a very explicit and nice embedding into $\mathrm{Spin}(N)$... [1]: https://i.sstatic.net/76pP3.jpg
