I am not an expert but the question:
"Does there exist a simply-connected closed Riemannian Ricci flat $n$-manifold with $SO(n)$-holonomy?"
is a well-known open problem. Note that Schwarzschild metric is a complete Ricci flat metric on $S^2\times\mathbb R^2$ with holonomy $SO(4)$, so the issue is to produce compact examples; I personally think there should be many. The difficulty is that it is hard to solve Einstein equation on compact manifolds. If memory serves me, Berger's book "Panorama of Riemannian geometry" discusses this matter extensively.
I think it would be more accurate to say that the real reason why
Calabi-Yau, hyperkähler, $G_2$ and $\mathrm{Spin}(7)$ manifolds are of
interest in string theory is not their Ricci-flatness, but the fact
that they admit parallel spinor fields. Of course, in
positive-definite signature, existence of parallel spinor fields
implies Ricci-flatness, but the converse is still open for compact
riemannian manifolds, as discussed in this
question, and known to fail for noncompact manifolds as pointed
out in an answer to that question.
The similar question for lorentzian manifolds has a bit of history.
First of all, the holonomy principle states that a spin manifold
admits parallel spinor fields if and only if (the spin lift of) its
holonomy group is contained in the stabilizer subgroup of a nonzero
spinor. Some low-dimensional (i.e., $\leq 11$, the cases relevant to
string and M-theories) investigations (by Robert Bryant and myself,
independently) suggested that these subgroups are either of two types:
subgroups $G < \mathrm{Spin}(n) < \mathrm{Spin}(1,n)$, whence $G$ is
the ones corresponding to the cases 5-8 in the question, or else $G =
H \ltimes \mathbb{R}^{n-1}$, where $H < \mathrm{Spin}(n-1)$ is one of
the groups in cases 5-8 in the question. Thomas Leistner showed that
this persisted in the general case and, as Igor pointed out in his
answer, arrived at a classification of possible lorentzian holonomy
groups. Anton Galaev then constructed metrics with all the possible
holonomy groups, showing that they all arise. Their work is reviewed
in their
paper (MR2436228).
The basic difficulty in the indefinite-signature case is that the de
Rham decomposition theorem is modified. Recall that the de Rham
decomposition theorem states that if $(M,g)$ is a complete, connected
and simply connected positive-definite riemannian manifold and if the
holonomy group acts reducibly, then the manifold is a riemannian
product, whence it is enough to restrict to irreducible holonomy
representations. This is by no means a trivial problem, but is
tractable.
In contrast, in the indefinite signature situation, there is a
modification of this theorem due to Wu, which says that it is not
enough for the holonomy representation to be reducible, it has to be
nondegenerately reducible. This means that it is fully reducible
and the direct sums in the decomposition are orthogonal with respect
to the metric. This means that it is therefore not enough to restrict
oneself to irreducible holonomy representations. For example,
Bérard-Bergery and Ikemakhen proved that the only lorentzian holonomy
group acting irreducibly is $\mathrm{SO}_0(1,n)$ itself: namely, the
generic holonomy group.
It should be pointed out that in indefinite signature, the
integrability condition for the existence of parallel spinor fields is
not Ricci-flatness. Instead, it's that the image of the Ricci
operator $S: TM \to TM$, defined by $g(S(X),Y) = r(X,Y)$, with $r$ the
Ricci curvature, be isotropic. Hence if one is interested in
supersymmetric solutions of supergravity theories (without fluxes) one
is interested in Ricci-flat lorentzian manifolds (of the relevant
dimension) admitting parallel spinor fields. It is now not enough to
reduce the holonomy to the isotropy of a spinor, but the
Ricci-flatness equation must be imposed additionally.
Best Answer
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.