We present a simple and novel classifier-based preordering approach. Unlike existing preordering models, we train feature-rich discriminative classifiers that directly predict the target-side word order. Our approach combines the strengths of lexical reordering and syntactic preordering models by performing long-distance reorderings using the structure of the parse tree, while utilizing a discriminative model with a rich set of features, including lexical features. We present extensive experiments on 22 language pairs, including preordering into English from 7 other languages. We obtain improvements of up to 1.4 BLEU on language pairs in the WMT 2010 shared task. For languages from different families the improvements often exceed 2 BLEU. Many of these gains are also significant in human evaluations.