Before Henry VIII's time, I think European royals had a very very very hard time getting divorced. But those who found themselves in impossible marriages sometimes could, if they had what lawyers call ``good facts'' and the right connections, persuade the Catholic Church to grant them an annulment. But I often wonder how often those grounds were really flimsy and were granted for political favors or cold hard cash. Seems to me there have been royal scandals in connection with annulments but I can't think of any off the top of my head.
Anne of Cleves got some sort of an annulment, but that was after Henry exited the RC church. Didn't Henry VIII's sister Margaret get an annulment? And were Josephine and Napoleon divorced or did he manage to get an annulment?
I don't think royals or influencial men ever had troubles getting rid of an unwanted wife. There are scores of women who found themselves in a convent, because they had fallen out of favor.
As to regular folks: in the pre-registered times, I presume, men simply left and took another wife in another hamlet.
Women have always had a much harder time to leave a husband.