adb pull

You can get the adb tool either from the Android SDK or from the android-tools package on Fedora (which seems to have an older version). You also have to enable the USB debugging mode on your phone (it is the first option on the developer options in the system settings).

With it all set up, you can use "adb pull" to copy files (or whole directory trees) from the device, "adb push" to send a file to the device, and "adb shell ls /sdcard/" (yes, you can get a shell with just "adb shell") to see what is there to copy. What you used to see on the mass storage mode is under /sdcard on the phone.

Do not forget to disable USB debugging when you are done, since as you can see it is quite powerful (it can do much more; for instance, it can install packages directly from your computer, without asking for confirmation on the device).