I realised that Satipatthana was the 'Four Noble Truths Meditation' only after reading about Satipatthana as one single meditative process. As far as I know only Reginald Ray and Bodhipaksa have described this (which I talk about here). The idea that Satipatthana is the 'Four Noble Truths Meditation' seems completely new - I've seen no mention of it anywhere - so don't worry if you haven't heard of it. But if you haven't yet, it's worth absorbing this information into your meditation.
We can see how Satipatthana is a Four Noble Truths Meditation only after seeing how the four foundations of mindfulness are used as a single meditative process, rather than a helpful way to divide up mindfulness. Bodhipaksa describes this in terms of a process starting with mindfulness of body, in which we experience mindfulness of feeling, followed by experience of our emotional/mind response to feeling - such as craving. Then having seen this process we drop the craving response and just stay with our feeling. Ray mentions the idea of Satipatthana as 'one single meditative process' coming from the Tibetan Tantric Tradition of which he is part, and he describes it a bit differently.
Bodhipaksa describes this process primarily in terms of feeling to emotion, but we can flip it over and see it from emotion to feeling as well, which is how I describe it, and I think this is even more helpful. It's when we see this process from volition* to feeling that we can more easily see how this is a Four Noble Truths Meditation. This is because it shows us how our volition is conditioning our feeling (not just how feeling is conditioning our volition). By dropping unskilled volition we then liberate ourselves from the unsatisfactoriness (feeling), so it shows us the way to - and the existence of - liberation; the third and fourth noble truths.
In the four foundations of mindfulness we see each of the Four Noble Truths, in 'feeling' we have the truth of unsatisfactoriness, in 'volition' we have the cause of unsatisfactoriness, and in the fourth foundation we have 'dhamma' - the path and goal of Buddhism. By dropping the cause of unsatisfactoriness - our own volitional activity - we liberate ourselves from unsatisfactoriness. All this happens only after we can bring our mind into our body - the first foundation of mindfulness. We can directly experience this for ourselves if we use Satipatthana in this way.
In practical terms, by bringing our mind into our body we experience feeling (satisfaction and dissatisfaction), and underlying that feeling we experience the mental and physical volition that conditions it (for example the mental hindrances), and by dropping that activity - which we can view as wasted effort - we experience the reduction of that unsatisfying feeling. This leads directly to liberation from dissatisfaction. It's this reduction element that makes Satipatthana 'the direct path'. Experientially, for us this means bringing our mind into our body to enjoy deeply satisfying relaxation, at deeper and deeper levels. That feeling of satisfaction is our main guide into the meditation.
Give it a try, I've uploaded a guided meditation here, and let me know how it goes. I also describe it in my book, The Path of Relaxation, if you want to read more about it.
*(I prefer 'volition', which I think is better than 'emotion', because 'citta' is more than emotion. By volition I mean both mental and physical activity which is both consciously and unconsciously made)