The largest demonstration took place in the capital, where a large crowd rallied in front of the Constitutional Tribunal and then marched across town to the palace of the president, who has played a key role in the crisis.
Protesters at the front of the march carried a large banner saying, "let's bring back the constitutional order."
Protests, also staged in Poznan and Wroclaw, were organized to show support for the beleaguered court and to urge the conservative ruling party, Law and Justice, to roll back changes that have undermined the court's ability to act as a check on government power.
There have been frequent street protests in the past three months.
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Also today, the government, which remains popular with its conservative electorate, said that it still refuses to publish a ruling by the Constitutional Tribunal that struck down the legislative amendments passed in December that have blocked the court and plunged the nation into crisis.
Refusing to publish the ruling prevents it from becoming binding.
The announcement by government spokesman Rafal Bochenek indicates a resolution is still nowhere in sight.
Yesterday, the Venice Commission, an international human rights body, said democracy is threatened by government moves that have "crippled" the constitutional court.
It said refusing to publish the judgement would violate the rule of law and further deepen the crisis.
Bochenek said the commission's opinion would be sent to the parliament so all political sides could seek a resolution.
The crisis, which has caused concern in the European Union and the United States, appears to be the most serious since Poland threw off communism in 1989.
"Democracy is fine, very fine," Beata Kempa, a leading official in the government of Prime Minister Beata Szydlo, said recently.
"We don't send police with bullets against people. They are allowed to express their views here.