This is a difficult question to answer for its breadth; but also because it's always hard to simply say that people judged everything from the past as either true or false. Many judgements probably will fall somewhere in between. To provide a few pointers I'll focus on examples mostly regarding Spain and Spanish America in the 16th century.

As Anthony Grafton has argued, classical and biblical authors provided an important frame of reference for early modern humanists, and for (sometimes humanist) authors grappling with the “discovery” of the Americas. In his (programatically titled) “New Worlds Ancient Texts” he shows the great influence of scholars like Pliny and Herodotus on later authors in viewing and describing the world – especially since the authors of antiquity still had an “aura of freshness” due to their recent rediscovery. As Grafton's analysis is quite detailed I'll quote one example of his:

Peter Martyr, an Italian humanist who worked in Spain, published his spectacular account of the New World in 1516, using the primary evidence of Columbus and much additional evidence, from language to customs. He encouraged those who actually went to the New World, as he did not, to report back systematically. But he also used a vast range of classical authorities to frame his account. He compared New World parrots with those described by Pliny, the life of the islanders on Hispaniola to the Golden Age depicted by Virgil and Hesiod, the elusive cannibals of the Caribbean to the Thracians who went to Lesbos to impregnate the Amazons.

Here we can see with the first chronicler of Spanish America a new form of gathering “evidence” by focusing on accounts by eye-witnesses. At this point however, these accounts are still mediated by the ancient authorities' descriptions, making the unknown and incredible people, animals and plants (but also apparently monsters) more credible to European scholars and readers.

Let's turn to another example that comes closer to the medieval time frame you're asking about: the reconquista. I'm interested here in how this concept from medieval times was used effectively in early modernity and beyond as a legitimizing tool by Spanish elites. From an earlier answer of mine:

The 'Primera Crónica General' was written in the late 13th c., started under predecessors of Alfonso X. of Castile and continued under Alfonso. It was a project involving various scribes and translators, but due to the king's participation can be seen as expressing royal viewpoints that proved influential on later universal chronicles. The providential conception of history [...] is evident, e.g. when the Visigoths' defeat is portrayed as divine punishment for their kings' sins, while the “survival” of the Northern Christian realms is also seen as divine intervention in order to upkeep the “light of Christianity” in Iberia. The description of how God helped the first Asturian ruler at the mythical battle of Covadonga (in the 8th c.) can be seen as an important legitimating strategy for Christian conquests, linking Visigothic territories with the later Christian kingdoms.

This narrative would prove influential in the colonization of the Americas as well, with many symbols and tales of the reconquista being „reused“ during and after the conquista, equating Spanish Muslim and native American populations. To this myth could be added other examples from religious authors, including the continuing importance of apocalyptical descriptions in early modern Spain and America. This meant that the coming of apocalypse described in the bible was taken up in the early 16th c. (and at other times) to position Charles V. as a saviour figure fighting against Protestants and Ottomans, both equated with the anti-Christ. The Franciscans would again transpose such narratives to the Americas.

My point here was to show the continuing influence of both biblical and classical authorities in early modern times. While new forms of gathering knowledge gained importance, especially with increasing research on the Americas, at least early on they still worked within the paramaters set by the traditional canon. As the reconquista example shows, this canon could also include more recent and local writings, like medieval chronicles, which continued to have legitimating importance. Then again, as I mentioned in the beginning, it's hard to say whether earlier tales were “in general” accepted as true. Despite what I said before, we can also find cases of authors defying authorities or traditional narratives. One example of the top of my head would be the scholar Antonio de Nebrija, who wrote the first Castilian grammar – as a first step in battling what he decried as the decline of the traditional education of Latin in Europe. There are surely many more.