GrooveIP and RedPhone might do it, since they're actually VoIP services that just happen to use your phone number to make the connection. Just using a Google Voice number rather than giving people your actual phone number will help, since it makes the connection through Google and therefore obfuscates the connection log.+Tim Malone
For some free and easy options (on Android, at least): use RedPhone to encrypt confidential calls, TextSecure to encrypt text messages you send as well as those saved on your phone (since lots of other apps have access to them otherwise), and use WiFi instead of mobile broadband whenever possible to give yourself a bit more anonymity and make mining your data a bit more difficult (and also save you money if you don't have an unlimited data plan). When browsing the web, use browser extensions like AdBlock and Ghostery to prevent your activity from being tracked, and use secure http (https) connections whenever possible (which the EFF's HTTPS Everywhere extension can help you with).
Finally, sign up with a VPN service that doesn't keep activity logs. (I use and recommend Private Internet Access, but there are quite a few others.). They're not free, but can be very affordable. And they perform two very important functions. First, they encrypt all of your internet access so no one can snoop on you on an open WiFi network. Second and more importantly to this discussion, they conceal your identity and location when using any form of internet communication, such as some of those mentioned above. For example, I'm currently typing this on my phone in Alabama, but anyone trying to trace my location or identity through my IP address would think I'm currently in Texas. Meanwhile, my desktop is in Toronto. Later tonight, I'll be in London so I can watch some videos on the BBC's iPlayer. Using an anonymous VPN service is probably the single best thing you can do to protect your privacy on the internet, and is particularly important when traveling in countries that don't respect privacy and freedom of speech online, such as Iran, Saudi Arabia, China, and the U.S.
Use an anonymous VPN service, use the internet for all of your communications, use encryption whenever possible, and use some common sense when doing things like logging into websites that might provide personally-identifiable information, and you're effectively immune to any attempts by the NSA or others to track you or record your communications.