I truly have not decided what I want to do here, but this article has seriously made me consider dropping late designations all together (having, instead, simply completed work or missing work). If you are interested in weighing in, I ask that you read the article above in its entirety, and then put your opinion here. Moreover, please engage with one another's ideas to help flesh them (and my own thinking) out more fully.
I have a couple ideas in mind that you could also reflect on. First, I would take Dr. Hasinoff's suggestion that there still be due dates, as the course is meant to work at a particular pace--work coming in after a due date would not be penalized but I would require you to visit office hours for feedback. Second, I would also be willing to enter into a binding contract with any student who felt that they needed deadlines--firm, enforceable deadlines--to stay on track. I welcome any other suggestions you might have! (However, to be clear, I have not made any decisions about a change to the current policy as outlined in our syllabus.)