Despite the clunky interface, I was pleasantly surprised by its biking directions: not as good as Google, but not terrible either. (I was in NYC when I tried using it for awhile.) It's fully open-source.
Just plugged in some of the test endpoints I used with MapQuest. YourNavigation seems to have two bicycling modes, "bicycle" and "bicycle (routes").
The former seems to disregard bike lanes entirely and generates some pretty hazardous recommendations (including cycling down 5th Ave, or busy sections of Park Ave in Midtown).
The "bicycle (routes)" mode does better and seems to respect bike lanes, but not consistently. There are some endpoints where it still maneuvers you into dense traffic with no bike infrastructure, when a bike-laned street is nearby and takes only marginally longer. I just checked the OSM's dataset and the data is good - so it isn't a matter of someone inserting a bike lane where one doesn't actually exist.
The app I'm working on is for NYC's new bikeshare system... Still not an acceptable alternative unfortunately. I may just give up and use Google Maps in the iOS app, but the performance of Google's native map component is extremely, extremely bad.
CycleStreets (UK cycle-based journey planner) are on the verge of open-sourcing their code - as in, in the next day or two. It's a highly fine-tuned cycle planner based on OSM data; here in the UK, at least, the results are streets ahead of Google, which will merrily send you down busy, fast trunk roads with no cycle lane.
You could either take their code and run an instance for NYC, or smile sweetly at them and ask if they might host one themselves. Github link: https://github.com/cyclestreets
http://www.yournavigation.org/
Despite the clunky interface, I was pleasantly surprised by its biking directions: not as good as Google, but not terrible either. (I was in NYC when I tried using it for awhile.) It's fully open-source.