Create Custom Big Notifications

Update due to API changes:

From API 24 on, the Notification.Builder contains a setCustomBigContentView(RemoteViews)-method. Also the NotificationCompat.Builder (which is part of the support.v4 package) contains this method.
Please note, that the documentation for the NotificationCompat.Builder.setCustomBigContentView states:

Supply custom RemoteViews to use instead of the platform template in the expanded form. This will override the expanded layout that would otherwise be constructed by this Builder object. No-op on versions prior to JELLY_BEAN.

Therefore, this will also only work for API >= 16 (JELLY_BEAN).

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Original Answer

So after excessive google usage, I found this tutorial explaining how to use custom big layouts. The trick is not to use setStyle() but manually set the bigContentView field of the Notification after building it. Seems a bit hacky, but this is what I finally came up with:

notification_layout_big.xml:

<?xml version="1.0" encoding="utf-8"?>
<LinearLayout xmlns:android="http://schemas.android.com/apk/res/android"
    android:layout_width="match_parent"
    android:layout_height="100dp" <!-- This is where I manually define the height -->
    android:orientation="horizontal" >

    <!-- some more elements.. --> 
</LinearLayout>

Building Notification in code:

Notification foregroundNote;

RemoteViews bigView = new RemoteViews(getApplicationContext().getPackageName(),
    R.layout.notification_layout_big);

// bigView.setOnClickPendingIntent() etc..

Notification.Builder mNotifyBuilder = new Notification.Builder(this);
foregroundNote = mNotifyBuilder.setContentTitle("some string")
        .setContentText("Slide down on note to expand")
        .setSmallIcon(R.drawable.ic_stat_notify_white)
        .setLargeIcon(bigIcon)
        .build();

foregroundNote.bigContentView = bigView;

// now show notification..
NotificationManager mNotifyManager = (NotificationManager) getSystemService(Context.NOTIFICATION_SERVICE);
mNotifyManager.notify(1, foregroundNote);

Edit
As noted by chx101, this only works for API >= 16. I did not mention it in this answer, yet it was mentioned in the given linked tutorial above:

Expanded notifications were first introduced in Android 4.1 JellyBean [API 16].