The cascade - or weeping window - section will be shown at St George's Hall in Liverpool and Woodhorn Museum, Northumberland.
Yorkshire Sculpture Park will host the arch segment, also known as the wave.
The installation, which marked 100 years since the start of World War One, drew more than five million visitors.
Culture Secretary John Whittingdale said: "This is art at its most powerful and it is only right that everyone should have the chance to see them."
The installation included 888,246 ceramic poppies, with each representing each death in the British and Colonial forces between 1914 and 1918.
It was created by Derbyshire artist Paul Cummins and theatre designer Tom Piper, and was named Blood Swept Lands and Seas of Red after a line written by a soldier who died in Belgium.
A total of £9m was raised for six military service charities after most of the poppies, which were handmade in Derby, were sold to members of the public for £25 each.
Wave and the Weeping Window, which together have more than 10,000 ceramic poppies, were bought for the nation last year by the charities Backstage Trust and the Clore Duffield Foundation.
Other locations can bid next year to host the ceramic poppies in 2017 and 2018, before they are permanently homed at the Imperial War Museums in London and Manchester.
Wave: Yorkshire Sculpture Park, Wakefield, September 2015 - January 2016
Weeping Window
Woodhorn Museum, Northumberland, September - October 2015
St George's Hall, Liverpool, November 2015 - January 2016
Source: 14-18 Now
World War One Centenary

Silverwood's side head the Division One table by 41 points with five games remaining, the first of them at home to Somerset starting on Monday.
Only Notts, in 2005, have won the title following promotion the previous year.
"In our first year back up in Division One, to bring it home would be pretty special." Silverwood told BBC Essex.
Essex lifted the trophy six times in the days of a single-division Championship, most recently in 1992 when Graham Gooch was captain.
Now, current skipper Ryan ten Doeschate has the task to leading the team to their first silverware of any kind for seven years following the end of their involvement in this summer's T20 Blast.
Only two of Essex's last five matches are at home, and they will have to manage without England batsmen Alastair Cook and Tom Westley, their two leading run-scorers in the Championship, because of the current Test series against West Indies.
"We are in a great position, but there's still a lot of work to be done and we won't take anything for granted," said Silverwood.
"We have got five games of Championship cricket left and we can look at them with a lot of excitement. We'll go out there and give every ounce of energy, blood sweat and tears.
"It's the holy grail, and certainly in our dressing room and among the members, although it's great to win (other) competitions, we want to bring that one home."
Thousands of ceramic poppies used in the Tower of London installation are to go on display across the north of England, it has been announced.
[NEXT_CONCEPT]
Essex head coach Chris Silverwood has set his sights on "the holy grail" of winning the County Championship before their return to red-ball action.