The National Union of Journalists (NUJ) called the strike, which began at midnight, after failing to reach an agreement with BBC management.
The strike was called over the redeployment of 30 staff members facing compulsory redundancy.
The union said jobs were set to be axed across the corporation, including BBC Scotland, Five Live, the Asian Network and the World Service.
The corporation is cutting about 2,000 jobs over five years as part of its Delivering Quality First programme.
Most of the 2,000 redundancies are expected to occur as a result of natural wastage and a freeze on recruitment.
There has already been disruption to BBC radio and TV schedules today.
The Today programme on Radio 4 was dropped and BBC Breakfast was replaced with a 30-minute bulletin on the hour, followed by daytime TV programmes including Cash in the Attic and Escape to the Country.
NUJ general secretary Michelle Stanistreet said: "NUJ members across the BBC are taking action to defend jobs and quality journalism at the corporation.
"They are angry and frustrated at the poor decisions being taken at the top of the BBC – decisions that are leading to journalists being forced out of their jobs and quality journalism and programming compromised.
"Instead of making sure that the redeployment process works properly in all areas of the BBC, managers are prepared to waste public money on needless redundancies and sacrifice the livelihoods of experienced and talented journalists, at the same time as advertising other jobs externally.
"It's particularly disappointing that the BBC has failed to engage meaningfully in attempts to resolve this dispute – an abdication of responsibility for a public service broadcaster."
A BBC spokesperson apologised for the disruption to services and said the corporation was "disappointed" about the strike.
"Unfortunately, industrial action does not alter the fact that the BBC has significant savings targets and as a consequence may have to make a number of compulsory redundancies," the spokesperson said.
"We have made considerable progress in reducing the need for compulsory redundancies through volunteers, redeployment and cancelling vacant positions, and we will continue with these efforts."