What's so special about a 2D plane in graph theory? See, we can divide graphs into planar ones and non-planar ones. This seems to make a lot of sense until you find that this seems to only work in two dimensions.
I can't think of any graph that cannot have a projection in three-dimensional space without any lines intersecting. Obviously, there are non-planar (in the 2d sense) graphs, such as the Petersen graph, but AFAIK it is very easy to make a 3D Petersen graph without any lines intersecting.
What is so special about 2 dimensions that allows nontrivial planar and non-planar graphs? In one dimension all nontrivial graphs are non-"linear", and it seems all three-dimensional graphs are "spacear".
 A: There is a theorem of Dimension Theory which says that an $n$-dimensional thingie (where thingie = separable metric space) can always be embedded in $(2n+1)$-dimensional Euclidean space; some of them require the full $2n+1$ dimensions, others can be embedded in lower-dimensional Euclidean spaces. A (linear) graph is $1$-dimensional, so all graphs can be embedded in $3$D space, some in $2$D, and a very few in $1$D.
If you want to learn about dimension theory, I can recommend a very nice and readable book from 1948, Dimension Theory by Witold Hurewicz and Henry Wallman. The prerequisite is a course in general topology.
Getting back to graphs, someone might quibble that a "graph" with more than $2^{\aleph_0}$ vertices can't be embedded in $\mathbb R^3$. I assume that we're talking about finite or countable graphs, which are separable metric spaces. Actually any (simple) graph with at most continuum-many vertices has a straight-line embedding (edges are straight line segments) in $\mathbb R^3$. Namely, map the vertices of the graph to distinct points on the twisted cubic curve $x=t,y=t^2,z=t^3$; it's easy to see that the straight line segments with endpoints on that curve do not intersect one another. 
