This is my style and I find it works well, if it works for you? All the better:
Shoot less, talk more. Spend your time getting to know them and having them interact together. As you are relaxing and making new friends shoot some snaps of them interacting naturally together. I find that the best portraits come that way. I like to sit with the family either on a sofa or on chairs, etc grouped together across from me so that when I do see a shot they are all there. Then relax. Ask them to tell you about each other. Ask things that will evoke great memories, smiles and some great kidding and laughing. Like to sibling #1 "if you could name the #1 most embarrassing moment in sibling #2's life what would it be?" "tell me a bout sibling #3's senior prom date?"
The photos you will get might just amaze you!
If you aren't ready to give up the backdrop and posed stuff entirely that is just fine!!! Just open yourself up to what just might happen by accident...