Researchers in textual entailment have begun to consider inferences involving downward-entailing operators, an interesting and important class of lexical items that change the way inferences are made. Recent work proposed a method for learning English downward-entailing operators that requires access to a high-quality collection of negative polarity items (NPIs).  However, English is one of the very few languages for which such a list exists.  We propose the first approach that can be applied to the many languages for which there is no pre-existing hight-precision database of NPIs.  As a case study, we apply our method to Romanian and show that our method yields good results.  Also, we perform a cross-linguistic analysis that suggests interesting connections to some findings in linguistic typology.