It will be a criminal offence to have an air weapon without a licence or permit from 31 December 2016.
Anyone found guilty of the new offence could be fined or face up to two years in prison.
Owners can apply to Police Scotland for a new air weapon certificate from 1 July.
The three-week amnesty began on 23 May and will run until 12 June.
Ch Insp Mike Whitford, of Tayside's divisional coordination unit, said 261 weapons had been handed in so far.
He said: "Every one of these weapons has the potential to cause serious injury or death, in particular to children or young people if handled incorrectly.
"There will be many lying around in homes across Tayside unused and this is a great opportunity to get rid of them."
The weapons can be handed in to police headquarters in Dundee, Perth and Forfar.
Mr Whitford said: "I would encourage anyone who has such a weapon to consider if they are likely to apply for a certificate under the new legislation and if not, hand it in to one of designated stations.
"It will remove the risk from their family and other members of the public.
"If you are carrying a weapon to a police station, please keep it in a bag to avoid causing alarm."
The legislation was passed by MSPs in June 2015.
The Scottish government pledged to introduce the licensing scheme following the death of Glasgow toddler Andrew Morton, who was shot dead by an airgun in 2005.
More than 260 air weapons have been handed in to Tayside police stations ahead of the introduction new Scottish licensing laws.