My Note 3 came with Samsungs Pedometer software (S Health) to monitor the motion sensitive chip in my phone. It's OK. I do enjoy the feedback I get from it and find it useful. The caveat is that it's sensitive to motion beyond walking such as driving or mass transit. On the daily graph, it's easy to pickout those points and mentally subtract them from the total. I suspect the fitbit is just as overly sensitive.
There is pedometer software in itunes/Play store that you can try out and see what you think of the concept. Saves you the bluetooth and extra charging hassles. For my tastes, these are all pretty snoopy pieces of kit in that they collect data about you that they don't necessarily keep private. To that end, I have rooted my phone and installed Xprivacy which gives me improved control over what data can be collected and shared.