In "Predictably Irrational", they mention a study (don't have the book with me, can't be more specific) where a teacher assigns three projects, due at various points throughout the year. The students do relatively well.

In another class the same teacher assigns three projects which can be turned in whenever the students want. The students in this class do quite a bit worse than the first students because they're procrastinating and do all three projects the last week.

In yet another condition, students are allowed to turn in projects at any time during the year, but they're also allowed to voluntarily pre-commit to a due date at the beginning of the year (they fail the project if they don't have it in by their own due date). In this condition, the students who pre-committed did as well as the students in the first class, and the students who didn't pre-commit did as poorly as the students in the second class.

Based on that study, I predict your students would do better if you gave them assigned due dates.