Edit

The BottomSheet is now part of the android-support-library. See John Shelleys’ answer.

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Unfortunately there’s currently no “official” way on how to do this (at least none that I’m aware of).
Luckily there’s a library called “BottomSheet” (click) which mimics the look and feel of the BottomSheet and supports Android 2.1 and up.

In case of the Drive app, here’s how the code would look like with this library:

    new BottomSheet.Builder(this, R.style.BottomSheet_Dialog)
            .title("New")
            .grid() // <-- important part
            .sheet(R.menu.menu_bottom_sheet)
            .listener(new DialogInterface.OnClickListener() {
        @Override
        public void onClick(DialogInterface dialog, int which) {
            // TODO
        }
    }).show();

menu_bottom_sheet (basically a standard /res/menu/*.xml resource)

<menu xmlns:android="http://schemas.android.com/apk/res/android">
    <item
        android:id="@+id/folder"
        android:title="Folder"
        android:icon="@drawable/ic_action_folder" />
    <item
        android:id="@+id/upload"
        android:title="Upload"
        android:icon="@drawable/ic_action_file_upload" />
    <item
        android:id="@+id/scan"
        android:title="Scan"
        android:icon="@drawable/ic_action_camera_alt" />
</menu>

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Output looks like this:

Which, I think, comes pretty close to the original. If you’re not happy with the colors you can customize it – see this (click).