I do give them assigned due dates, actually; I just make it clear to everyone that I'll accept arbitrarily late work without penalty. I make a show of collecting papers on the date they're due. About 80--90% of the students turn in their work on the due date.

Adding to the incentive scheme here is the fact that the classwork builds on earlier work, so if they just procrastinate for a few weeks, they'll be in a pretty dire situation. If they have to go see a funeral or a concert, or they've been really busy some week with exams or a big project for another class, or whatever, then my method gives them enough flexibility for that, but the great majority of people do not procrastinate the way you might predict.

I guess you could call this system "obviously fictitious due dates". Everybody knows that the due dates are made of handwaving and rainbows, but that doesn't make them much less effective. And this way I don't have to listen to any damn excuses, and my students don't have to think of any; I just smile and recommend catching up before next week.