The protesters say they also want constitutional and judicial reforms.
The demonstrations were organized by more than 20 political parties and nongovernmental organizations.
They come ahead of elections on April 9 to fill vacant parliament seats in three constituencies.
The demonstrations follow a decision by the country's highest court to allow a controversial businessman to run in one of the parliamentary by-elections. Election officials had initially barred Rysbek Akmatbaev from running, a decision that triggered protests by his supporters.