Operators affected by the Aslef union walkouts include Northern, which runs services from York to Leeds, and TransPennine, whose services run from York to cities including Leeds, Manchester and Liverpool, and to Malton and Scarborough.
Unions involved in disputes must reballot their members every six months on continuing with industrial action.
Aslef members at Northern voted by 93 per cent to continue striking and by 95 per cent on TransPennine Express trains.
The turnout for ballots across all operators was at least 70 per cent.
Mick Whelan, general secretary of Aslef said: “These results show – yet again – a clear rejection by train drivers of the ridiculous offer put to us in April last year by the Rail Delivery Group on behalf of the train operating companies with whom we are in dispute.
“The RDG knew the offer would be rejected because we had told them that a land grab for all the terms and conditions we have negotiated over the years would be unacceptable.
“Since then our members have voted, time and again, for strikes. That’s why Mark Harper, the Transport Secretary, is being disingenuous when he says that offer should have been put to members.
“Drivers obviously wouldn’t vote for industrial action, again and again and again, if they thought that was a good offer. They don’t.
“That offer was dead in the water in April last year – and I think Mr Harper knows that.”
Mr Whelan said the union remained open and willing to talk about a revised offer.
“That’s why we are asking the Secretary of State for Transport, or the Rail Minister Huw Merriman, to come and meet us,” he said.
