In the capital Nairobi and other cities, many shops remain open.
At least 24 people have been killed in violence since the 8 August election, the Kenya National Commission on Human Rights has said.
Mr Odinga said the poll was rigged to give President Uhuru Kenyatta victory, and called for a "day of mourning for the fallen patriots".
However, Kenyans on Twitter have been using the hashtag #TurudiKaziniChallenge (Swahili for "Let's return to work" ) to urge a return to normalcy following last week's disruptions.
Mr Kenyatta repeated his appeal for peace.
"At the end of the day we are all Kenyans, we don't need to fight one  another, we don't need to destroy each other's property, we don't need to take  life," he said in a statement.
"Kenyans have said that the election is behind them, the majority have  returned to work," he added.
Young people have been burning tyres in Kibera, a Nairobi slum where Mr Odinga has strong support, reports the BBC's Tomi Oladipo from the scene.
However, some of Mr Odinga's supporters in Kibera have opened their shops, saying that while they agree with him that the result was rigged, they cannot afford to lose business, our reporter adds.
Businesses are open as usual in the coastal city of Mombasa.
In the western city of Kisumu, a stronghold of Mr Odinga, there has been a mixed response to the strike, with shops open in some areas and shut in others.
The official results gave President Kenyatta about 54% of the vote, and Mr Odinga 45%.
"This is a failed regime that is resorting to killing people instead of addressing the real issue. The vote was stolen. There's no secret about that," Mr Odinga told his supporters on Sunday, as he called for a strike.
Foreign observers have declared the poll free and fair.
European Union foreign affairs chief Federica Mogherini and former UN Secretary General Kofi Annan are among those who have urged Mr Odinga to seek redress through the courts - something the opposition coalition says is not an option.

The poster features an officer and a message to parents asking them not to use the threat of the police when disciplining their children.
Durham Constabulary wants children to see officers as people who can help rather than simply law enforcers.
And the force's poster has been shared more than 47,000 times and liked by almost 35,000 people on Facebook.
A force spokesman said: "Children should be taught from an early age that if they feel like are in danger and they need help, they can approach an officer who will reassure them."
However, the poster has divided opinion on Facebook.
Jeff Rowse said: "Telling them they will go to jail if they are bad is wrong how? Surely that is the major reason for having a police force? And jails?"
In response Paul Parry said: "Telling a three-year-old that if he doesn't eat his dinner the policeman will come and take him to jail isn't true and will only serve to instil fear into that child.
"The police shouldn't be used to reinforce bad parenting."
Others were more taken with the officer, with several people asking for him to be moved to their local force or saying they would happily go to him for help.

Police and other emergency services had responded to reports of a man in difficulty at about 13:45 on Saturday.
The incident took place as a yacht racing event was being held in the area.
The man was brought ashore but died at the scene. Police said no further details would be released until he has been formally identified.
A spokesman said there were no apparent suspicious circumstances.
A report will be submitted to the procurator fiscal.
Many Kenyans have ignored opposition leader Raila Odinga's call for a strike over disputed elections.
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A poster urging parents not to use the police to scare their children has been seen more than 3.5 million times.
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A 69-year-old man has died after his boat capsized in the Findhorn Bay area of Moray.