You are right that they shouldn't affect the minds of modern Ireland but unfortunately the reality is that the Plantation of Ulster has had long lasting effects on Irish society. In my opinion it is undoubtedly the single most important factor in the troubles as it lay the foundations of what would happen in the future. Here's an extract from an essay by John Darby who was an expert on the conflict, he can summarise it better than I can!
The Plantation of Ulster was unique among Irish plantations in that it set out to attract colonists of all classes from England, Scotland and Wales by generous offers of land. Essentially it sought to transplant a society to Ireland. The native Irish remained, but were initially excluded from the towns built by the Planters, and banished to the mountains and bogs on the margins of the land they had previously owned. The sum of the Plantation of Ulster was the introduction of a foreign community, which spoke a different language, represented an alien culture and way of life, including a new type of land tenure and management. In addition, most of the newcomers were Protestant by religion, while the native Irish were Catholic. So the broad outlines of the current conflict in Northern Ireland had been sketched out within fifty years of the plantation: the same territory was occupied by two hostile groups, one believing the land had been usurped and the other believing that their tenure was constantly under threat of rebellion. They often lived in separate quarters. They identified their differences as religious and cultural as well as territorial.