I think they absolutely had a grudge. Yes, it’s true, the Dornish are quite open about paramours, but you’ll note that paramours tend to be of a lower class than the wife or husband of the person in question, and thus not a social threat. Lyanna was a Stark of Winterfell and thus of sufficient standing to marry a prince and totally inappropriate to be a mistress. If Rhaegar elopes with her, everyone’s first thought is going to be that he’s setting his wife aside. There’s also an expectation of some discretion – and Rhaegar had openly snubbed his wife (who had just been told could not have another child) in favor of the new hottie.
Secondly, there’s a good deal of evidence that the Martells especially were pretty pissed off – they only sent 10,000 men to save the capitol, Aerys was resorting to open hostage-taking to ensure loyalty, and the surviving Martells tend to speak of Rhaegar with some tension.