A Kakeya set is a subset of F^n, where F is a finite field of q elements, that contains a line in every direction. In this paper we show that every Kakeya set is of size at least (C_n)*(q^{n-1}), where C_n depends only on n. This improves the previously best lower bound for general n of ~ q^{4n/7} due to Mockenhaupt and Tao (Duke Math. J. 2004).