The strike is aimed at forcing Prime Minister Sheikh Hasina to quit and form a caretaker government made up of people from outside political parties to oversee an election due by early next year. The shutdown began Sunday morning and is to continue until Tuesday night.
An activist from the main opposition Bangladesh Nationalist Party was killed in the southwestern district of Faridpur after security officials opened fire on stone-throwing protesters, local police official Abul Kalam said, adding that at least six people were injured.
He said police opened fire after opposition activists attacked them in the area, which is 64 kilometers (40 miles) southwest of the capital, Dhaka.
A backer of the ruling Awami League party was hacked to death, allegedly by opposition activists, in Jessore, 135 kilometers (85 miles) west of Dhaka, local police chief Abdus Salek said by phone.
In the southern district of Pirojpur, another ruling party supporter was hacked to death by activists from the country's largest Islamic party, Jamaat-e-Islami, police official M.A. Karim said.
Violence in the northern district of Pabna left a Jamaat-e-Islami member dead, the Bengali-language Prothom Alo daily reported.
Dozens of opposition activists were arrested across the country, Mirza Fakhrul Islam Alamgir, a senior opposition leader, told reporters in Dhaka.
The dispute centers around who will oversee an election the government has to hold within the next three months. A system of caretaker governments taking people from outside the parties has been used for 15 years, but the government scrapped it after the Supreme Court ruled that the system contradicts the constitution.
The opposition, led by former Prime Minister Khaelda Zia, has demanded that the system be restored and has threatened to boycott the election. The government rejects the demand, and earlier this month proposed forming an all-party government instead.
In a phone conversation Saturday night, Hasina invited Zia, her archrival, to dinner at her official residence on Monday to discuss the issue and requested a stop to the general strike. Zia refused, but said she would consider the invitation after the strike expires Tuesday.
Bangladesh, a parliamentary democracy, has been alternately ruled by Hasina and Zia since 1991. But the issue of peaceful transfers of power has remained a major challenge.
Television stations said the violence had left scores hurt across the country since Saturday, with explosions of homemade bombs reported in Dhaka and elsewhere, including bombs thrown at the official homes of two senior justices and the offices of four TV stations and newspapers.
Several vehicles were torched in Dhaka on Sunday, and Channel 24 TV station said opposition activists attacked a train and set fire to it, injuring at least 20 people in Joypurhat district, about 210 kilometers (130 miles) north of Dhaka.
On Friday, similar violence left at least six people dead across the country.