How do I "shake" an Android device within the Android emulator to bring up the dev menu to debug my React Native app

Within your app in the Android Emulator press Command + M on macOS or Ctrl + M on Linux and Windows.

With a React Native running in the emulator,
Press ctrl+m (for Linux, I suppose it's the same for Windows and ⌘+m for Mac OS X) or run the following in terminal:

adb shell input keyevent 82

If you're using the new emulator that comes with Android Studio 2.0, the keyboard shortcut for the menu key is now Cmd+M, just like in Genymotion.

Alternatively, you can always send a menu button press using adb in a terminal:

adb shell input keyevent KEYCODE_MENU

Also note that the menu button shortcut isn't a strict requirement, it's just the default behavior provided by the ReactActivity Java class (which is used by default if you created your project with react-native init). Here's the relevant code from onKeyUp in ReactActivity.java:

if (keyCode == KeyEvent.KEYCODE_MENU) {
  mReactInstanceManager.showDevOptionsDialog();
  return true;
}

If you're adding React Native to an existing app (documentation here) and you aren't using ReactActivity, you'll need to hook the menu button up in a similar way. You can also call ReactInstanceManager.showDevOptionsDialog through any other mechanism. For example, in an app I'm working on, I added a dev-only Action Bar menu item that brings up the menu, since I find that more convenient than shaking the device when working on a physical device.