The soldiers pulled an elaborately decorated chariot containing the urn from the Grand Palace through the heart of old Bangkok.
The ceremony began at the Grand Palace on Friday, with King Bhumibol Adulyadej, Thailand's monarch, leading the royal family and more than 100 specially chosen monks in a mourning ceremony.
The king lit candles and incense, as his wife Queen Sirikit and their children looked on.
The cremation itself will be on Saturday afternoon.
The ceremony will provide a brief break from the protest-fueled political sparring that has gripped Thailand since anti-government protests began in May.
In the worst political violence in the country in 16 years, two protesters were killed and more than 470 injured when the anti-government protesters and riot police clashed on October 7.
Protesters have vowed to suspend their anti-government vitriol for the duration of Galyani's funeral.
"We will cease making political remarks during the period to pay respect to the late princess," Chamlong Srimuang, one of the protest leaders, said.
The protesters accuse the government of acting as proxy for Thaksin Shinawatra, the former prime minister who was ousted in a 2006 military coup.
'Act of merit'
Somchai Wongsawat, Thailand's current prime minister and Thaksin's brother-in-law, called on "all sides to put their differences aside" in order to mourn the princess.
"Everyone should mourn and perform good deeds as an act of merit to honor the late princess," he said.
The princess, who was born in London on May 6, 1923, was known as modest and self-effacing, telling an interviewer in 2000: "I don't like gala dinners. They're boring."
She was also noted for her interest in theater and classical music, a taste she cultivated when, like the king, she was educated in Switzerland.
Galyani is survived by a daughter and a grandson.