In general, for dimensions $n>5$ we don't have topological obstructions to the existence of Einstein metrics (like as Hitchin-Thorpe inequality when $n=4$).
However, there exist compact homogeneous Riemannian manifolds with no $G$-invariant Einstein metrics. In this case, one works with cosets $M=G/K$ of a compact Lie group $G\subset {\rm Isom}(M)$ and the Einstein equation $Ric = c \cdot g$ is written with respect to a $G$-invariant Riemannian metric $g$ on $M=G/K$. For such a metric the Einstein equation reduces to a polynomial system and positive real solutions correspond to invariant Einstein metrics.
In particular, let $\frak{g}=\frak{k}\oplus\frak{m}$ be a reductive decomposition for $G/K$ (such an orthogonal decomposition always exists for a homogeneous Riemannian manifold $(M=G/K, g)$). For simplicity, let us assume that tangent space $\frak{m}=T_{o}G/K$ decomposes into $q$ pairwise inequivalent isotropy summands (i.e., irreducible $Ad(K)$-submodules)
$ \frak{m}= \frak{m}_1 + \frak{m}_2 + ..... + \frak{m}_q. $
Then any $G$-invariant Riemannian metric is given by
$ g = < , > = x_1\cdot (-B)|_{\frak{m}_1}+\cdots+x_s\cdot (-B)|_{\frak{m}_q} $
for some positive real numbers $x_1, x_2, ... x_q$, where $-B$ denotes the negative of the Killing form of $\frak{g}$ (restricted on $\frak{m}$).
Homogeneous Einstein metrics are real solutions of the system $\{r_1-r_2=0, ...., r_{q-1}-r_{q}=0\}$ where $r_{i}$ are the components of the invariant Ricci tensor on any isotropy summand. These are given by
\begin{equation}\label{ricc} r_{k}=\frac{1}{2x_{k}}+\frac{1}{4d_{k}}\sum_{i, j}\frac{x_{k}}{x_{i}x_{j}}[ijk]-\frac{1}{2d_{k}}\sum_{i, j}\frac{x_{j}}{x_{k}x_{i}}[kij], \qquad (k=1, \ldots, q). \end{equation} Here $[ijk]$ are the structure constants of $G/K$. Their computation is usually non-trivial. For particular classes of homogeneous spaces (e.g. isotropy irreducible), they are related with the Casimir operator of $\frak{g}$. They are non-zero only for these triples $\frak{m}_i$, $\frak{m}_j$, $\frak{m}_k$, for which
$B([\frak{m}_i, \frak{m}_j], \frak{m}_k)\neq 0$.
Positive real solutions of the above system may exist or not. This system, and the study of homogeneous Einstein metrics becomes more complicated if some of the isotropy summands are equivalent each other.
It turns out that here are homogeneous spaces with NO invariant Einstein metrics. For example, Wang and Ziller, by applying the variational approach of homogeneous Einstein metrics on compact homogeneous spaces, they proved that the 12-dimensional space $SU(4)/SU(2)$ does not admit any homogeneous Einstein metric.
see: M. Wang and W. Ziller: Existence and non-existence of homogeneous Einstein metrics}, Invent.~Math.~84 (1986) 177--194.
This space seems to be the lowest dimensional example of a compact homogeneous manifold with no invariant Einstein metrics. In particular, from a recent work of B"ohm and Kerr we have the more general result:
Theorem: Any simply connected compact homogeneous Einstein manifold admits at least an invariant Einstein metric.
B"ohm and M. Kerr: Low-dimensional homogeneous Einstein manifolds, Trans. Amer. Math. Soc. 358 (4) (2005) 1455--1468.
Other interesting examples of compact homogeneous spaces with no invariant Einstein metrics have been presented also in
J-S. Park and Y. Sakane: Invariant Einstein metrics on certain homogeneous spaces, Tokyo J. Math. 20 (1) (1997) 51--61.
More general arguments about the existence of homogeneous Einstein metrics (which are based on the topology of compact homogeneous spaces and applications of variational analysis), can be found in the following articles:
C. B"ohm, M. Wang and W. Ziller: A variational approach for homogeneous Einstein metrics}, Geom. Funct. Anal. 14 (2004) (4) 681-733.
C. LeBrun and M. Wang (editors): Surveys in Differential Geometry} Volume VI Essays on Einstein Manifolds, International Press, 1999.
C. B"ohm: Homogeneous Einstein metrics and simplicial complexes, J. Diff. Geom. 67 (2004) 79-165.
We mention that less are know about the problem of non-homogeneous Einstein metrics on homogeneous spaces (see the work of Page, or Bohm for the existence of non homogeneous Einstein metrics.)
For non-compact homogeneous manifolds (solvmanifolds, etc) we refer the reader to Heber's work and the references therein, although a lot of progress has been made in the last decade in this case too (see the works of Laurent, Tamaru, and others).
J. Heber: Non-compact homogeneous Einstein space, Invent.~ Math.~133 (1998) 279-352.