The military-installed government has banned protests by Brotherhood members demanding the reinstatement of deposed president Mohamed Morsi, after listing the movement as a terrorist organisation on Wednesday.
The three people were killed as opponents of the Islamists clashed with protesters in several cities, the interior ministry said.
Riot police fired tear gas at stone-throwing protesters in several cities, swiftly clamping down on rallies after the midday today prayers, the usual time for Islamist demonstrations.
The interior ministry said police arrested 265 "Brotherhood elements" in the clashes.
"Legal measures are underway against the Brotherhood elements in accordance with the cabinet's listing of the Muslim Brotherhood as a terrorist organisation," the ministry said in a statement.
The Brotherhood's designation as a terrorist group carries harsh penalties, with the group's leaders facing possible death sentences and protesters looking at up to five years in prison.
Smoke rose from Al-Azhar university's student dormitory in Cairo as police fired tear gas against protesters pelting them with rocks from inside the building, the latest in a series of student demonstrations that have turned violent.
Protesters wounded several policemen across the country today and torched police vehicles, the interior ministry said.
