The march comes amid rising anger over the draft charter and decrees issued by Mohammed Mursi giving himself sweeping powers and immunise the Shura council from dissolution.
The protesters dubbed the marches "the Final Warning".
The protesters, who gathered close to Ittihadiya Palace, could not reach the palace after security cordoned off all streets leading up there with barbed wire.
The protesters chanted, "We want bread, freedom and the to bring down Constituent Assembly" and "Down with the rule of the [Muslim Brotherhood] Supreme Guide."
Eleven privately-owned newspapers stopped publication today in protest against the limits on freedom of expression in Egypt's draft constitution.
The one day blackout was the sharpest strike yet in a new push by liberal and secular groups to defeat the draft charter, which was approved on Friday by an Islamist-dominated assembly despite the boycotts and objections of almost all non-Islamist delegates.
The decree draft that one purpose of the news media is to uphold public morality and the "true nature of the Egyptian family," and it requires authorisation to operate a television station or a Web site.
It is Egypt's worst political crisis since the ouster nearly two years ago of authoritarian president Hosni Mubarak.