Thousands of staff walked out for three days in June as part of the largest rail strike for more than 30 years, causing widespread disruption.
Further strikes will now take place in July and August, starting next week, with rail journeys in Scarborough and across the Yorkshire Coast affected.
The strikes have been organised for Wednesday July 27, Thursday August 18 and Saturday August 20 with all passengers urged to seek alternative routes for their journey.
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Mick Lynch, RMT's General Secretary, said: "The offer from Network Rail represents a real terms pay cut for our members and the paltry sum is conditional on RMT members agreeing to drastic changes in their working lives."
Aslef has announced that train drivers at eight railway companies will strike on Saturday July 30 after talks over pay broke down.
However, the walkout of 5,500 staff does not currently affect Northern or TransPennine Express, which provides train services in the region.
Northern, which operates some routes out of Scarborough, said it "expects significant disruption to travel" and has issued a 'do not travel' warning for Wednesday July 27.
Tricia Williams, chief operating officer at Northern, said: "We've taken on board learnings from the previous RMT strikes last month and have done everything we can to offer a skeleton service on the parts of our network where we're able to - but our advice remains not to travel."
It also said some services on the following day may be affected by the strikes.
TransPennine Express said the strikes will cause "significant disruption" to services and urged passengers to plan ahead and seek alternative transport.
The Transport Salaried Staffs' Association has also voted to join the industrial action. Unions must, by law, give 14 days' notice of a strike taking place.
Network Rail said only around half of Britain's rail network will be open on July 27, with "very limited services" running on lines that will be open, and urged passengers who must travel to plan ahead.
Trains between Scarborough and Hull were cancelled on Monday July 18 after the railway line had become blocked "due to the severe weather", amid nationwide fears of track buckling in the extreme heat.
Services between Scarborough and Sheffield were also cancelled, with disruption between Filey and York. A rare 'do not travel' warning was issued for Tuesday July 19.
Network Rail said that railway lines in direct sunshine can be 20°C hotter than the air temperature and because rails are made from steel, they expand as they get hotter, and can start to curve – which is known as 'buckling'.
Last month, rail strikes caused severe disruption to train journeys to and from Scarborough.
Scarborough Railway Station, Seamer and Malton were all completely closed with no services calling on strike days.
The industrial action fell on the same day as the Armed Forces Day national event, which attracted more than 100,000 visitors to the town and caused widespread journey disruption.
The Rail, Maritime and Transport union announced the strikes after talks over pay and redundancies fell through.
Services in the area were extremely limited, with half of all rail lines in the country closed.