The question as to whether cultures evolve in a manner analogous to that of genetic evolution can be addressed by attempting to reconstruct population histories using cultural data. As others have argued, this can only succeed if cultures are isolated enough to maintain and pass on a central core of traditions that can be modified over time. In this study we used a set of cultural data (canoe design traits from Polynesia) to look for the kinds of patterns and relationships normally found in population genetic studies. After developing new techniques to accommodate the peculiarities of cultural data, we were able to infer an ancestral region (Fiji) and a sequence of cultural origins for these Polynesian societies. In addition, we found evidence of cultural exchange, migration and a serial founder effect. Results were stronger when analyses were based on functional traits (presumably subject to natural selection and convergence) rather than symbolic or stylistic traits (probably subject to cultural selection for rapid divergence). These patterns strongly suggest that cultural evolution, while clearly affected by cultural exchange, is also subject to some of the same processes and constraints as genetic evolution.