The Middle Ages is so influential that there is a special branch of study, medievalism, which looks at romantic versions of the Middle Ages.
The romantic Middle Ages really started in the nineteenth century. During the Middle Ages themselves, of course, when everyone thought they lived "now" rather than in some "middle" period between antiquity and modernity, writers might wax nostalgic for some imaginary past, but they didn't call this past the Middle Ages.
The Italian Renaissance (which is actually late medieval Italy in disguise) invented the idea of a "middle" period between antiquity and them, who were of course "modern." The Protestant Reformation of the sixteenth century deliberately broke with medieval religion. By the time we reach the eighteenth-century Enlightenment, the Middle Ages was labeled an age of superstition and darkness.
So what happened to stop this slide of the Middle Ages into the "middle-evil" period one may see on student exams? (I have to tell mine that this is not a funny joke.)
Basically it was the Industrial Revolution, that got off the ground in England toward the end of the eighteenth century and really spread in the nineteenth. While radically dropping the price of manufactured goods and making all sorts of new inventions possible, industrialization also created massive pollution, a sharp separation between the artisan and the manufactured product, and people working in factories where they were treated as interchangeable cogs.
In response, there was a new yearning for a time before all this happened, and the time chosen was the Middle Ages. In this age of chivalry and faith, it was assumed, the leaders were brave and honorable, beauty was cherished, the sturdy yeoman farmer took pride in his labor while knowing his place, and life was full of passion and rich meaning. Well, maybe.
Sir Walter Scott, author of Ivanhoe, really started this romanticism. He was a contemporary of Jane Austen, although from looking at his stories of knightly derring-do and hers of country gentry trying to get their daughters married one might never think so. (He admired her very much, though noting that one one but he himself could do as good a job at what he called "the big bow-wow scene." I think you had to be there.)
The most famous promoter of a romanticized Middle Ages was Richard Wagner (d. 1883), whose opera "The Ring of the Nibelung" is based on two medieval epics, the "Volsung Saga" and the "Nibelungenlied." These two medieval epics, though coming out of the same long oral tradition, where tales of Merovingian-era kings were mixed with the old story of Sigurd the Dragonslayer until they were almost unrecognizable, both rejected major aspects of the other's story. The Saga is set in a pagan universe, where the gods are very active and Sigurd has a hot affair with Odin's daughter Brunhild. The Nibelungenlied is set in a Christian universe, where Siegfried had never had a relationship with Brunhild, queen of Iceland. (He does get to slay a dragon in both, however.)
This did not bother Wagner (pictured above), who picked and chose what he liked from both, to create a series of operas of epic proportions, where industrialization was bad, heroes were half-way to being gods, and a search for wealth was very evil.
In the US, medieval architecture became very popular. The simplicity of white New England churches was replaced by neo-Gothic (or neo-Romanesque) styles. Office buildings and apartment buildings were often built to have a medieval look (think Richardsonian Romanesque). Ironically, these buildings used modern industrial techniques.