In adding syntax to statistical MT, there is a tradeoff between taking advantage of linguistic analysis, versus allowing the model to exploit linguistically unmotivated mappings learned from parallel training data. A number of previous efforts have tackled this tradeoff by starting with a commitment to linguistically motivated analyses and then finding appropriate ways to soften that commitment. We present an approach that explores the tradeoff from the other direction, starting with a context-free translation model learned directly from aligned parallel text, and then adding soft constituent-level constraints based on parses of the source language. We obtain substantial improvements in performance for translation from Chinese and Arabic to English.