We have previously proposed localized graph frames (LGFs) as flexible, intuitive, and computationally efficient tools for graph signal processing. We use these frames to introduce a new class of graph filters that consist of a concatenation of LGF analysis, scalar weight multiplication, and LGF synthesis. These LGF filters have low implementation complexity and are inherently nonstationary, which is a key advantage compared to conventional graph filters. We provide an in-depth analysis of LGF filters that reveals that their properties are effectively determined by the coefficient-domain weight function. A denoising experiment confirms the superiority of LGF filters in terms of SNR improvement and complexity.