My comment seemed to get jumbled, so here is an expanded version:
In 3-d, an ancient complete Ricci flow (so, in particular a shrinking soliton) will have non-negative sectional curvature. This follows from a result of Chen, http://arxiv.org/pdf/0706.3081.pdf, Corollary 2.4. (If the manifold has controlled geometry at infinity, this follows from earlier results like Hamilton-Ivey pinching).
However, in higher dimensions, Hamilton-Ivey style pinching estimates do not work. (*). Indeed, as your question asks, there are shrinkers with non-positive Ricci curvature. For example, the solitons constructed by Feldman-Ilmanen-Knopf in http://projecteuclid.org/euclid.jdg/1090511686 do not have non-negative Ricci curvature.
An interesting question is whether or not these can arise as singularity models for a compact Ricci flow, and this was answered in the affirmative by Maximo: http://math.stanford.edu/~maximo/Maximo%20-%20On%20the%20blow-up%20of%20four%20dimensional%20Ricci%20flow%20singularities.pdf. An interesting consequence of his work is that positive Ricci curvature is not preserved along the flow in higher dimensions (EDIT: as remarked the answer by @BewSMA, the lack of preservation of Ricci curvature under the flow was already observed).
EDIT: I'll just point out that the above paper of Chen proves that a shrinker must have non-negative scalar curvature (see Corollary 2.5).
(*) In spite of this aspect of the theory failing, there are curvature conditions which are preserved under the flow. For example, see http://www.ams.org/journals/jams/2009-22-01/S0894-0347-08-00613-9/S0894-0347-08-00613-9.pdf