Google has been increasingly transitioning new features in Android to a more closed model. Whether you're talking about music playback, search, or even the dialer, Google's updated apps have features not included in the Android Open Source Project. The developers of OmniROM are looking to make the handy features of the new Google Dialer available without the proprietary bits, so they're working on an integrated phone number directory without Google.
This feature is currently in development, so you can't download a version of OmniROM that includes it just yet. When it's ready for the public, the new OmniROM dialer will plug into OpenStreetMap to pull in local search results when you enter keywords, just like the new Google Dialer. Google doesn't get any data, and you only need to make your approximate location available. This will probably come as a relief to those with a bad case of Google-phobia.
The new OmniROM dialer will also include the option to use the Google Places API, just for completeness sake. Well, and there are still things Google is better at than any of the more open alternatives. The integrated phone number search will roll out in a future build of OmniROM, but the devs didn't offer any timeline.