Moscow is braced for what
the opposition claims will be the biggest demonstration in Russia for 20
years.
Smaller rallies are taking place in cities across the country.
Protesters allege there was widespread fraud in Sunday's polls - though the ruling United Russia party saw its share of the vote fall sharply.
Hundreds of people have been arrested during anti-Putin protests over the past week, mainly in Moscow and St Petersburg.
At least 50,000 police and riot troops have been deployed in Moscow ahead of Saturday's protests.
The opposition says it is hoping for a turnout of 30,000 in the capital in the demonstration dubbed "For Fair Elections", due to begin at 14:00 (10:00 GMT).
Protests have already begun elsewhere, with several hundred marching in Vladivostok, seven timezones to the east of Moscow.
Protest relocated
The CA's Daniel Sandford in Moscow
says in the past week, the city has resembled a police state rather than a
democracy.If the protests come even close to expectations, they will shake the 12-year-long political domination of Prime Minister Vladimir Putin, he says.
The authorities permitted demonstrations to take place in specific locations in certain cities after negotiations with opposition leaders.
In Moscow, the two sides reached a deal by which authorities would allow a high-turnout if the rally was relocated from downtown Revolution Square to Bolotnaya Square, a narrow island in the Moscow River.
In St Petersburg, 13,000 people have pledged on the social networking site Vkontakte to take part in protests, with another 20,000 saying they might take part.
Authorities have granted permission for a demonstration in one location, but
say protests anywhere else will be illegal and will be dealt with. The official results of the elections to Russia's Duma showed that the ruling United Russia party saw its share of the vote fall from 64% to 49%, though it remained easily the biggest party.
But there is a widespread view, fuelled by mobile phone videos and accounts on internet social networking sites, that there was wholesale election fraud and that Mr Putin's party cheated its way to victory, our correspondent says.
On Friday, the presidential Council for Human Rights advising Mr Medvedev said the reports of vote-rigging were of deep concern, and that the elections should be rerun if they were confirmed.
However the council has no power to order a fresh ballot, correspondents say.
Earlier this week, security experts said attempts had been made to counter online dissent in Russia, with hijacked PCs being used to drown out online chat on Twitter.
Analysis of the many pro-Kremlin messages posted to some discussions suggested they were sent by machines, according to security firm Trend Micro.
Momentum
These are the most significant street protests against Mr Putin since he
took power, our correspondent says - but at this point they are not drawing the
big numbers they would need to really put the Kremlin in trouble. It will be a question of seeing whether the momentum builds and spreads from the metropolitan middle classes.
Even so, our correspondent adds, it is an extraordinary thing to witness Mr Putin under fire like this.
Mr Putin, who was president between 2000 and 2008, remains widely predicted to win a presidential election in March.
On Thursday, he blamed the US for stoking the recent unrest, after Secretary of State Hilary Clinton expressed reservations over the poll.
The prime minister said Mrs Clinton's remarks had "set the tone for some opposition activists".
