Do new gravitational effects emerge in higher dimensions? Effects like gravitational waves and the curvature surrounding black holes do not occur in spacetimes with one time-like coordinate and two space-like coordinates. This is because the Einstein Field Equations fully constrain the Riemann tensor in fewer than four dimensions.
Are there other interesting effects that exist in four spacial dimensions plus a time-like dimension that vanish in our (3,1) universe? Or is the  qualitative difference between (2,1) and (3,1) solely due to the transition between a fully-constrained Riemann tensor and one with freedom?
 A: I) For starters, for higher-dimensional GR with $n\geq 5$ spacetime dimensions, an event horizon (which always has codimension 2) needs not be homotopic to a sphere $S^{n-2}$. E.g. for $n=5$, there are also black rings. 
II) On the other hand, low-dimensional GR with $n\leq 3$ spacetime dimensions 


*

*has identically zero Weyl curvature tensor;

*is a topological field theory with no local physical propagating fields, i.e no gravitational waves. 
A: A physical way to see that there are no gravitational waves in spacetime dimension $d~=~3$ is with the degrees of freedom of a wave. An electromagnetic or other gauge field has an electric $\vec E$ and magnetic field $\vec B$. In a sourceless region These fields are orthogonal in an electromagnetic or gauge field wave. Yet if you have only two spatial dimensions you do not have enough dimensions for the wave to propagate. The old right hand rule for the $\vec E$ and $\vec B$ fields and the direction or propagation $\vec k$ means you have insufficient number of spatial dimensions for waves. This means you have static field configurations, which is a topological field theory. 
The extends to gravitational waves as well which require in a weak field limit the perturbing metric elements $h_{++}$ and $h_{\times\times}$ for two polarization directions. This means you have traceless field tensors $E_{ij}$ and $B_{ij}$ for the electric and magnetic analogues of the gravitational field in the spatial manifold embeded in spacetime. Again if you have only two spatial dimensions you "run out of dimensions" for wave progagation. This is manifested more mathematically in the vanishing of the Weyl tensor. There is though a parallel tensor with conformal structure called the Cotton tensor. The Weyl tensor defines the conformal properties of spacetime, and the Cotton tensor does the same for lower dimensional spacetimes.
For dimensions higher than $4$ we can think about black holes with Gauss theory in a Newtonian setting. If we have any spatial surface $\Sigma^n$ with dimension $n$ larger than $2$ the gravitational field inside a Gaussian surface $S$ enclosing a region $V~\subset~\Sigma^n$ we have
$$
\int_S {\bf F}\cdot da~=~\int_V\nabla\cdot {\bf F}dV~=~4\pi G\rho,
$$
for $\rho$ the mass density. For three dimension the area enclosing the source is a $2$-sphere and we get Newtonian gravity. In general we get a form of this field that appears as
$$
{\bf F}~=~-\frac{k}{r^{n-1}}.
$$
and the potential that gives the field ${\bf F}~=~-\nabla U$ has a form 
$$
U~=~-\frac{k}{r^{n-2}}.
$$
The metric elements of the Schwarzschild metric $g_{tt}~=$ $g_{rr}^{-1}~=~(1~-~2m/r$ become modified accordingly with $g_{tt}~=~(1~-~c/r^{n-2})$. This is admittedly rather heuristic and a more rigorous derivation is needed. 
A: There are lots and lots of qualitatively new things that emerge in higher dimensions.  As QMechanic mentions, event horizons don't need to be topologically spherical. It's also known that generic initial conditions can lead to naked singularities, so the cosmic censorship hypothesis fails. I believe that black holes with hair also exist in higher dimensions.
