  We study Fano schemes $F_k(X)$ for complete intersections $X$ in a projective
toric variety $Y\subset \mathbb{P}^n$. Our strategy is to decompose $F_k(X)$
into closed subschemes based on the irreducible decomposition of $F_k(Y)$ as
studied by Ilten and Zotine. We define the expected dimension for these
subschemes, which always gives a lower bound on the actual dimension. Under
additional assumptions, we show that these subschemes are non-empty and smooth
of the expected dimension. Using tools from intersection theory, we can apply
these results to count the number of linear subspaces in $X$ when the expected
dimension of $F_k(X)$ is zero.
