You could just make a function that calls itself (if needed after some amount of time like after some number of milliseconds)

I know that the creative coding frameworks in C++ use a similar setup(), update(), draw() construct...

Cinder: http://libcinder.org/docs/v0.8.4/anatomypage.html

oF: http://www.openframeworks.cc/tutorials/developers/001_how_openFrameworks_works.html

draw() isn't part of the language, but as amnon.owed wrote, it is part of the graphical framework.
To add to the list, the Löve game framework, built around the wonderfully simple Lua language, uses a mechanism very similar to the Processing one, with love.load() called at the start, love.update(dt) called to update the world (dt is delta time, the time spent since the last call, useful to ensure regular moves if the call is delayed) and love.draw() called to display the world state.
Like in Processing, there can be some other callback functions, to handle keyboard and mouse (and joystick), etc.

So, basically, it is a general mechanism, not necessarily seen in every framework (JavaFX 1.0 used something different), but quite natural.

I see. it's probably not the ideal forum to ask how to achieve something similar in C# (I'm making an app for Windows 8), although asimes' idea sounds like it could work. I've looked at the Lua language briefly before, it does look simple and seems to be used in x-platform mobile development a bit this days too.

To answer my own question re C#, I've discovered that putting your code inside the 'Tick' of a Dispatcher Timer is equivalent to using the draw() function in processing.