  In this paper we consider $L^p$ Liouville type theorems for harmonic
functions on gradient Ricci solitons. In particular, assume that $(M,g)$ is a
gradient shrinking or steady K\"ahler-Ricci soliton, then we prove that any
harmonic function $u$ on $M$ with $\nabla u\in L^p(M)$ for some $p>1$ is a
constant function.
