Yes, there are many such results. Conway-Gordon, Sachs in the 80s proved that any map $K_6 \to R^3$ contains two disjoint linked traingles. Robertson-Seymour-Thomas proved found the family of minors that characterizes such property. Lovasz-Schrijver proved that this is equivalent to having Colin de Verderie invariant larger than 4 and the projection on the null space of the Colin de Verderie matrix is a linkless embedding (in the case the null space is of dimension four or less, I forget if this is a theorem or a conjecture?)
There are many papers saying things like, for your favorite Link invariant there is a numnber $n$, such that for any embedding $K_n \to R^3$ one can find a link with nontrivial your favorite invariant. I don't remember the references now, maybe google "ramsey theory for links" or something like that. ($K_n$ is the complete graph on $n$ vertices).
From a more geometrical point of view, here are two things you can do:
One is to look at metric properties. For this look up Kolmogorov-Borodin and the recent paper by Guth and Gromov. Actually expanders were discovered for this reason.
The alternative is to think about the linear structure, namely you can ask whether there are affine subspaces of the ambient space intersecting many of the edges for any embedding. In a recent joint paper with Boris Bukh we called this "space crossings". Because if the affine flat that intersects your edges is of dimension 0 this is precisely a crossing. We investigated the "space crossing numbers" of graphs in $R^3$, but our techniques generalize to graphs in $R^d$. The first result in this direction was Zivaljevic's who proved that $K_{6,6} \to R^3$ has non zero space crossing number. Our main result is an analogue of the classical crossing number inequality which almost implies it.