I propose and demonstrate the use of the recirculating delayed self-heterodyne (DSH) method for measuring FM noise power spectral densities (PSDs), which are the most fundamental measure characterizing the spectral purity of laser sources. By analyzing the DSH beat signals with 1, 10, and 160 km delays, the FM noise PSD of a narrow-linewidth fiber laser is evaluated for Fourier frequency range between 10 Hz and 100 kHz, which exhibits flicker noise as the dominant contribution.