  Let $X$ be an affine, simplicial toric variety over a field. Let $G_0$ denote
the Grothendieck group of coherent sheaves on a Noetherian scheme and let
$F^1G_0$ denote the first step of the filtration on $G_0$ by codimension of
support. Then $G_0(X)\cong\mathbb{Z}\oplus F^1G_0(X)$ and $F^1G_0(X)$ is a
finite abelian group. In dimension 2, we show that $F^1G_0(X)$ is a finite
cyclic group and determine its order. In dimension 3, $F^1G_0(X)$ is determined
up to a group extension of the Chow group $A^1(X)$ by the Chow group $A^2(X)$.
We determine the order of the Chow group $A^1(X)$ in this case. A conjecture on
the orders of $A^1(X)$ and $A^2(X)$ is formulated for all dimensions.
