Any riemannian manifold with holonony contained in $SU(n) \subset SO(2n)$, $Sp(n) \subset SO(4n)$, $G_2 \subset SO(7)$ and $Spin(7) \subset SO(8)$ are Ricci-flat. There are plenty of non-flat examples; e.g., those with holonomy precisely those groups.

In the Lorentzian setting, you could consider a subclass of lorentzian symmetric spaces with solvable transvection group, the so-called Cahen-Wallach spacetimes. They are pp-waves with metric given in local coordinates by $$ 2 dudv + \sum_{i=1}^{n-2} dx_i^2 + \sum_{i,j=1}^{n-2} A_{ij} x^i x^j du^2 $$ where $A_{ij}$ are the entries of a symmetric matrix. If $A$ is traceless, the metric is Ricci flat, but if $A \neq 0$ then it is non-flat.

Added (in response to the comment)

The riemannian result is classical. I learnt this from a book by Simon Salamon, “Riemannian Geometry and Holonomy Groups”, but there are some more recent lecture notes you might find useful: http://arxiv.org/abs/1206.3170. Concerning the lorentzian result, it is a simple calculation to determine the Riemann and Ricci tensors of the metric I wrote down. The metrics were introduced in the paper “Lorentzian symmetric spaces” by Michel Cahen and Nolan Wallach, Bull. Am. Math. Soc. 76 (1970), 585–591.

All riemannian manifolds with holonomy contained in $SU(n) \subset SO(2n)$, $Sp(n) \subset SO(4n)$, $G_2 \subset SO(7)$ and $Spin(7) \subset SO(8)$ are Ricci-flat. There are plenty of non-flat examples; e.g., those with holonomy precisely those groups.

In the Lorentzian setting, you could consider a subclass of lorentzian symmetric spaces with solvable transvection group, the so-called Cahen-Wallach spacetimes. They are pp-waves with metric given in local coordinates by $$ 2 dudv + \sum_{i=1}^{n-2} dx_i^2 + \sum_{i,j=1}^{n-2} A_{ij} x^i x^j du^2 $$ where $A_{ij}$ are the entries of a symmetric matrix. If $A$ is traceless, the metric is Ricci-flat, but if $A \neq 0$ then it is non-flat.

Added (in response to the comment)

The riemannian result is classical. I learnt this from a book by Simon Salamon, “Riemannian Geometry and Holonomy Groups”, but there are some more recent lecture notes you might find useful: http://arxiv.org/abs/1206.3170. Concerning the lorentzian result, it is a simple calculation to determine the Riemann and Ricci tensors of the metric I wrote down. The metrics were introduced in the paper “Lorentzian symmetric spaces” by Michel Cahen and Nolan Wallach, Bull. Am. Math. Soc. 76 (1970), 585–591.

Any riemannian manifold with holonony contained in $SU(n) \subset SO(2n)$, $Sp(n) \subset SO(4n)$, $G_2 \subset SO(7)$ and $Spin(7) \subset SO(8)$ are Ricci-flat. There are plenty of non-flat examples; e.g., those with holonomy precisely those groups.

In the Lorentzian setting, you could consider a subclass of lorentzian symmetric spaces with solvable transvection group, the so-called Cahen-Wallach spacetimes. They are pp-waves with metric given in local coordinates by $$ 2 dudv + \sum_{i=1}^{n-2} dx_i^2 + \sum_{i,j=1}^{n-2} A_{ij} x^i x^j du^2 $$ where $A_{ij}$ are the entries of a symmetric matrix. If $A$ is traceless, the metric is Ricci flat, but if $A \neq 0$ then it is non-flat.

Any riemannian manifold with holonony contained in $SU(n) \subset SO(2n)$, $Sp(n) \subset SO(4n)$, $G_2 \subset SO(7)$ and $Spin(7) \subset SO(8)$ are Ricci-flat. There are plenty of non-flat examples; e.g., those with holonomy precisely those groups.

In the Lorentzian setting, you could consider a subclass of lorentzian symmetric spaces with solvable transvection group, the so-called Cahen-Wallach spacetimes. They are pp-waves with metric given in local coordinates by $$ 2 dudv + \sum_{i=1}^{n-2} dx_i^2 + \sum_{i,j=1}^{n-2} A_{ij} x^i x^j du^2 $$ where $A_{ij}$ are the entries of a symmetric matrix. If $A$ is traceless, the metric is Ricci flat, but if $A \neq 0$ then it is non-flat.

Added (in response to the comment)

The riemannian result is classical. I learnt this from a book by Simon Salamon, “Riemannian Geometry and Holonomy Groups”, but there are some more recent lecture notes you might find useful: http://arxiv.org/abs/1206.3170. Concerning the lorentzian result, it is a simple calculation to determine the Riemann and Ricci tensors of the metric I wrote down. The metrics were introduced in the paper “Lorentzian symmetric spaces” by Michel Cahen and Nolan Wallach, Bull. Am. Math. Soc. 76 (1970), 585–591.