We perform a systematic search for rotationally invariant cosmological solutions to toy matrix models. These models correspond to the bosonic sector of Lorentzian Ishibashi, Kawai, Kitazawa and Tsuchiya (IKKT)-type matrix models in dimensions d less than ten, specifically d = 3 and d = 5. After taking a continuum (or commutative) limit they yield d - 1 dimensional Poisson manifolds. The manifolds have a Lorentzian induced metric which can be associated with closed, open, or static space-times. For d = 3, we obtain recursion relations from which it is possible to generate rotationally invariant matrix solutions which yield open universes in the continuum limit. Specific examples of matrix solutions have also been found which are associated with closed and static two-dimensional space-times in the continuum limit. The solutions provide for a resolution of cosmological singularities, at least within the context of the toy matrix models. The commutative limit reveals other desirable features, such as a solution describing a smooth transition from an initial inflation to a noninflationary era. Many of the d = 3 solutions have analogues in higher dimensions. The case of d = 5, in particular, has the potential for yielding realistic four-dimensional cosmologies in the continuum limit. We find four-dimensional de Sitter dS(4) or anti-de Sitter AdS(4) solutions when a totally antisymmetric term is included in the matrix action. A nontrivial Poisson structure is attached to these manifolds which represents the lowest order effect of noncommutativity. For the case of AdS(4), we find one particular limit where the lowest order noncommutativity vanishes at the boundary, but not in the interior.