In general, for dimensions n>5 we dont have  topological obstrucions  to Einstein metric ( (like as Hitchin-Thorpe inequality when n=4).
However, there are a lot of known compact homogeneous Riemannian manifolds with no $G$-invariant
Einstein metrics. In this case we work on quotient $G/K$ of  a compact Lie group and we consider the Einstein equation $Ric = c \cdot g$  for an invariant metric on M.
For such a metric the Einstein equation reduces to a polynomial system and positive real solution correspond to invariant Einstein metrics.

For example if G/K has a reductive decomposition $\frak{g}=\frak{k}\oplus\frak{m}$ and we assume that the tanfent 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 $x_1, x_2, ...  x_q$.  Here $-B$ is the negative of the Killing form. (the induced inner product on the tangent space).

Homogeneous Einstein metrics are real soulutions 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.
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, s).
 \end{equation}
where $[ijk]$ are the structure constants of $G/K$.


Such a system may have or not have positive real soultions, and thus there are (compact, but also non compact) homogeneous spaces with no invariant Einstein metrics. 
For example, Wang and Ziller by applying the variational approach to homogeneous spaces proved that the 12-dimensional space $SU(4)/SU(2)$ doe not admit any homogeneous Einstein metrics.

(see: M. Wang and W. Ziller: 
    Existence and non-excistence 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 know that :

 **Theorem:**  any simply connected compact homogeneous Einstein manifod admts 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 homogeneosu spaces with no invariant Einstein metrics were given by Sakane and Park : 

  J-S. Park and Y. Sakane:
   Invariant Einstein metrics on certain homogeneous spaces,
  Tokyo J. Math.  20   (1) (1997) 51--61.)

More general arguments based on topology of compact homogeneous spaces and the variational analysis, about the existence of homogeneous EInstein metrics are presented in the articles

 1)
  C. B\"ohm, M. Wang and W. Ziller:
   A variational approach for homogeneous Einstein metrics}, Geom. Funct. Anal. 14 (2004) (4) 681-733.

2) C. LeBrun and M. Wang (editors):
   Surveys in Differential Geometry} Volume VI Essays on Einstein Manifolds,
 International Press, 1999.

3)
 C. B\"ohm:
     Homogeneous Einstein metrics and simplicial complexes, 
   J. Diff. Geom. 67 (2004) 79-165.


 However, we mention that for the problem of non-homogeneous Einstein  metrics on homogeneous spaces, less are knwon (see the work of Page, or Bohm for the existence of non homogeneous Einstein metrics.)
For non compact homogeneous manifolds (solvmanifolds, nilmanifolds, etc) we refer the reader to Heber's work and the refernces therin, although a lot of progress has been made in the last decade in this case too.

J. Heber:
  Noncompact homogeneous Einstein space, 
Invent.~ Math.~133  (1998)   279-352.