dhenry has skipped over a few things here.
The standard included libraries initialize the timers, and in particular sets up Timer 0 so that it "ticks" every 1.024 mS, which is then used by the millis(), micros() and delay() function calls.
So if you were to totally bypass the IDE you would have to do that yourself. The init() function in the Arduino library is the one that does that. The file (wiring.c) also includes an interrupt handler to count those ticks.
However the answers above are correct. The Arduino "sketch" is treated as a C++ program (after a bit of text munging), so anything you can do in C++ you can do on the Arduino. Some compiler options disable some things, however, such as exception handling.
You are free to create new "tabs" in the IDE, and name your files xxx.c or xxx.cpp, giving you the ability to code in "straight" C or C++. Again, within the constraints of what compiler options are enabled, and what the included libraries allow. For example, the standard library does not let you write to stdout, because such a concept doesn't really exist on the microprocessor.