In theorem 1.12 on page 5 of http://www.math.utah.edu/dc/tilting.pdf, which states:
Given a bounded acyclic quiver $(Q,R)$, the K-theory of it's representations is given by $\mathbb{Z}^{Q_0}$
why is the acyclic hypothesis is needed? At the beginning of the proof of the theorem, Craw states that acyclicity ensure the simple objects in the category of representations is exactly the collection of simple vertices, but I do not understand where acyclicity fits into the picture.
The simplest quiver that isn't acyclic is the quiver with a single vertex and a single loop from that vertex to itself. A representation of this quiver is a vector space $V$ and an endomorphism $T : V \to V$, and when $V$ is finite-dimensional over $\mathbb{C}$, for example, these are classified by Jordan normal form. In particular, the simple objects are given by the $1$-dimensional representations where $T$ acts by some scalar; there are uncountably many of them, and in particular there isn't just one.
More generally, any time you have a cycle, the conjugacy class (so e.g. the characteristic polynomial) of the composite of every element in that cycle is an invariant of the quiver representation, and will generally take uncountably many values.