How do I prevent Android taking a screenshot when my app goes to the background?

Be careful about using WindowManager.LayoutParams.FLAG_SECURE, on some devices (verified on Samsung Galaxy ACE, e.g. GT-S5830) this will make the view scrambled. Looks like a Samsung specific bug. I recommend the following:

if(android.os.Build.VERSION.SDK_INT >= android.os.Build.VERSION_CODES.HONEYCOMB) {
    getWindow().setFlags(WindowManager.LayoutParams.FLAG_SECURE, WindowManager.LayoutParams.FLAG_SECURE);
}

This is what a scrambled screen looks like:

This is working properly on ICS Samsung phones though, so I'm assuming problem is isolated to Gingerbread devices (or older).

Try FLAG_SECURE:

public class FlagSecureTestActivity extends Activity {
  @Override
  public void onCreate(Bundle savedInstanceState) {
    super.onCreate(savedInstanceState);

    getWindow().setFlags(WindowManager.LayoutParams.FLAG_SECURE,
                         WindowManager.LayoutParams.FLAG_SECURE);

    setContentView(R.layout.main);
  }
}

This definitely secures against manual screenshots and automatic screenshots from the ICS recent-tasks history. It also secures against screen recording (e.g., apps using the media projection APIs).

UPDATE: it also secures against Now On Tap or other assistants on Android 6.0; they will not get access to the details of widgets and containers in your UI if the user brings up the assistant.

UPDATE #2: however, not everything in the activity will be protected. Any pop-up windows — Dialog, Spinner, AutoCompleteTextView, action bar overflow, etc. — will be insecure. You can fix the Dialog problem by calling getWindow() on it and setting FLAG_SECURE. The rest... gets tricky. See this blog post for more.