Many people are aversive to the language of "God" and "radical acceptance" removes this as a criterion. Accepting life on life's terms is another catch-phrase that mirrors radical acceptances..
But that's really what it is about. Accepting life on life's terms. It doesn't mean agreeing with it. It doesn't mean liking it. It doesn't even mean you won't kick and scream initially about the unfairness, or lack of justice. Radical acceptance is for YOU and not for the situation or the other person. Marsha Linehan says "suffering is optional, pain is not." Being human is to have and experience pain. Suffering occurs when pain continues based on the lack of accepting the pain. It takes practice, it takes being mindful, it takes courage.
It also takes practice. A lot of practice.
It can be very helpful for a person to begin by learning to "sit with themselves". Slowing down, reminding yourself that in this moment you're alive, you're breathing. The thoughts you have are exactly them - thoughts. The feelings you have are exactly that... feelings.. Work on noticing and describing them, non-judgmentally. Sit with your breath and open your palms to the world. Let whatever needs to come in move through you and notice. You can feel any amount of pain, any amount of annoyance - notice your thoughts.
Mindfulness is simple but not easy and it takes practice. Digging your heels in and "cutting your nose off to spite your face" is what causes misery. You can't change pain but in time, you can learn to sit with yourself and notice without judgment.
This is a good article explaining more on "radical acceptance" and defines exactly what it is and how to use it.