The 24-hour walk-out involves staff in Germany, Sweden, Ireland, Belgium and the Netherlands. About 50,000 passengers are understood to have been told of cancellations on 400 flights.
The action is the largest in a series of strikes over pay and conditions.
Ryanair says it has made every effort to resolve the dispute.
The company says 85% of its scheduled flights will operate as normal on Saturday and the majority of customers affected have been given places on other Ryanair flights.
The Irish airline, Europe's largest low-cost carrier, averted widespread strikes before Christmas by agreeing to recognise unions for the first time in its 30-year history.
But there have been protests ever since over the negotiating of collective labour agreements.
About 300 Ryanair flights were cancelled in July when cabin crews in Belgium, Portugal and Spain went on strike for 48 hours.
Emily Dutton was due to fly to Gothenburg in Sweden for a wedding, with her boyfriend Craig.
But the "once in a lifetime experience" of being at her best friend Justine's wedding seems impossible now, after Ryanair cancelled her flight with two days' notice.
The 27-year-old project manager from Letchworth was due to fly from Stansted Airport, and was told there were no alternative flights for the rest of the day.
"The only reason I found this out was through a reminder email about checking in being sent to my email account.
"Had I not checked, I would have arrived at the airport only to find my flight being cancelled.
"I spent about four hours [yesterday] calling them and trying to find my own travel options to see how to get to Sweden.
"They couldn't find us any alternative flights until Sunday. Great news, the wedding is on Saturday so that was absolutely no use."
She tried to find any other flights to Sweden from other UK airports, and even asked if she could fly to Copenhagen in Denmark, and get a train to Gothenburg instead.
"My only options were to get a refund or to go down the route - which I have done - to claim compensation."
The airline in recent days announced the cancellations of 250 flights in and out Germany, 104 to and from Belgium and another 42 affecting Sweden and Ireland.
A Dutch court on Thursday evening rejected a case from Ryanair seeking to block pilots in the Netherlands from joining the strike, affecting about 22 flights. But at about the same time Ryanair issued a statement saying there would be no cancellations.
The unions want the contracts of Ryanair employees to be governed by the laws of the nation where they are based, not by Irish legislation.
Ryanair has described the strike action as "regrettable and unjustified", claiming its pilots are paid more than other budget airlines.
It said it has taken every step to minimise the disruption, adding: "The majority of customers affected have already been re-accommodated on another Ryanair flight."
The airline has maintained that strikes qualify as "extraordinary circumstances" and are therefore exempt from EU compensation rules.
But the Civil Aviation Authority says that, in the case of strikes by an airline's employees, the airline is responsible for compensation.
Consumer group Which? said affected passengers should lodge a claim with the airline.
Which? travel editor Rory Boland said the dispute resolution service used by Ryanair had already confirmed it would uphold a previous European ruling that crew strikes are not usually considered "extraordinary circumstances".
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