UXD Researcher Laura Bowden recently gave a presentation at my school about this exact topic in which she suggested several methods.
Some general principles for mobile usability testing she gave were:
a. "When testing a native app, focus is on the device TYPE."
b. "When testing the mobile web, and/or responsive/adaptive design, focus is on the FORM FACTOR."
Before testing on a mobile device, make sure that users want to be able to perform the task you're testing on a mobile device (as opposed to in person, on a laptop, or calling on a phone).
If you want to be able to record what users do on the screen, there are mobile camera mounts for usability testing.
There are apps (such as RecordDisplay for iPad and iPhone) that let you record the device screen to a videofile, which isn't quite as good at recording the user's every motion, but at least gives you a record of what pages they went to on your site and how quickly.
If you're willing to jailbreak the phone, Cydia has applications that allow you to log the user's taps or swipes.
If none of the other options work for you, you could try using an online mobile-device simulator on a desktop computer, such as the iPhone4 Simulator.
The key thing to remember about mobile devices is that people use them on-the-go, often in particular favorite settings. The most effective testing methods will thus travel with the user to their preferred location, where there may be a lot of noise causing a reduction in attention span.