Aristotle, the primary source of medieval epistemologists, is famously obscure on the subject. The well-known (Metaphysics 1011b25) sounds fairly modern in assigning truth to statements:"To say of what is that it is not, or of what is not that it is, is false, while to say of what is that it is, and of what is not that it is not, is true". However, there are contrary passages in Metaphysics, discussed in Crivelli's book Aristotle on Truth to the effect that "the strictest truth" resides "in things themselves":
"In Θ10, Aristotle seems to imply that ‘being in the strictest sense true’ and ‘being in the strictest sense false’ hold of objects, but in E4, he appears to say that objects cannot be true or false: ‘falsehood and truth are not in objects… but in thought’ (1027b25-7)."
Charles argues in a recent Aristotle on Truth–Bearers that careful reading of the text supports statements/beliefs as the primary truth bearers, but the more nuanced interpretation that was developed by Aquinas, and summarized in his formula veritas est adaequatio rei et intellectus, incorporates a metaphysical aspect, and adequates the "truth of being" to the "truth of the intellect". The root of it is his addition of esse (act of existence) to the Aristotelian matter/form ontology, and the idea that truth is grounded more in the thing's esse than in its essence. It is by the intellect assimilating to the esse of the thing that the adequation is accomplished.
There is a complex theory of how this assimilation happens, involving both passive and active (agent) intellect, among other cognitive powers, but, long story short, it is the esse of things that causes the truth in the intellect. Does it mean that the truth has causal powers? Not quite, for esse is not the truth as such. The truth applies primarily to the intellect (in accordance with Aristotle's dominant paradigm), secondarily to propositions, and only analogically to things, insofar as the things cause the primary truth about themselves in the intellect. The relativism is easily ruled out by means expected of a scholastic thinker: both being and knowing have a common cause that ensures their adequation, God, the prime cause of it all.
Here is Wippel's explanation of Aquinas's position in Truth in Thomas Aquinas:
"In developing this point Thomas also notes that a thing is said to be true because it is naturally fitted by reason of its external appearances to produce a correct understanding of itself in the intellect. And other things are referred to as "false" because they are naturally suited by reason of their external appearances to produce a false understanding of themselves. Consider, for instance, something which appears to be gold but is not."
[...] "In sum, Thomas has singled out three levels of truth in this discussion. Truth in the full and complete sense is assigned to the intellect insofar as the intellect's grasp of a thing corresponds to that thing as it is in itself. Truth is then assigned to things themselves, but only analogically, because of their capacity to produce truth in the intellect. Finally, truth in the fullest and most perfect sense is assigned to God because he causes both the being of all other things and the acts of knowing of all other intellects. At the same time, it is clear from this discussion that Thomas reserves truth in the full and primary sense for it insofar as it is present in the intellect. It is here, apparently, that it best meets its description as an adequation of the intellect and the thing known."
Petruzella in Aristotle, Aquinas, and the Convertibility of Truth and Being gives a similar account. It should be noted, however, that Aquinas's position, like his introduction of esse, was rather unique among the scholastics. With others, the position on truth is closely related to the position on the nature of universals. For realists, like Duns Scotus, the truth resides in things quite literally, through their objective common natures imported into the intellect. On the other hand, for nominalists, like Ockham, it only applies to statements and propositions, see e.g. Ockham's Theory of Signification by Boehner.
A modern position, somewhat reminiscent of Aquinas's theory, is Davidson's, which is that the relation between reality and our concepts is not correspondence but causation, see an interesting take by Atkins in Can Perceptions Justify Beliefs? And it illustrates what happens when God is subtracted from the picture - without transcendent adequation realism becomes problematic. This is the major theme of criticism of Davidson in McDowell's neo-Aristotelian Mind and World.