The first explosion struck outside a bar a few streets away from Yangon's City Hall. Police quickly sealed off both blast sites.
The AFP news agency reported a Yangon police official as saying: "It was a bomb blast.
"A taxi nearby was hit and glass was broken nearby, but no one was injured."
No group immediately claimed responsibility for the bombings.
Myanmar's military rulers have not accused anyone of the bombing, though in the past similar incidents have been blamed on ethnic groups, including the Karen National Union, which is battling the military.
In January, one woman was injured in an explosion at Yangon's railway station, while earlier this month in Naypyidaw, the country's new capital, a woman was killed in a similar bombing at a train station.
Myanmar's generals say the new constitution will give some ethnic groups more autonomy and pave the way for multi-party elections in 2010, but pro-democracy activists say the charter simply entrenches military rule.
The country underwent extreme political turmoil in September last year when the government crushed peaceful pro-democracy demonstrations.
The UN estimates at least 31 people were killed and thousands more detained during the crackdown.