The Nearby Young Moving Groups (NYMGs) of stars are ideal for the study of evolution circumstellar disks in which planets may form because their ages range from a few Myr to about 100 Myr, about the same as the interval over which planets are thought to form. Their stars are distributed over large regions of the sky. Hence, the Wide Field Infrared Survey Explorer (WISE) which scanned the entire sky in four bands from 3.4 to 22.1 mu provides a database well-suited for the study of members of the NYMGs, particularly those identified after the eras of the IRAS and Spitzer observatories. We report our study of the stars in the epsilon and eta Cha, TW Hya, beta Pic, Tuc-Hor, and AB Dor NYMGs. The WISE Preliminary Release Source Catalog, which covers 57% of the sky, contains data for 64% of the stars in our search lists. WISE detected the 11.6 and 22.1 mu emission of all the previously known disks except for the coldest one, AU Mic. WISE detected no disks in the Tuc-Hor and AB Dor groups, the two oldest in our sample; the frequency of disks detected by WISE decreases rapidly with age of the group. WISE detected a circumstellar disk associated with 2M J0820-8003, a pre-main sequence star with episodic accretion in the ~ 6 Myr old eta Cha cluster. The inner radius of the disk extends close to the star, ~0.02 AU and its luminosity is about a tenth that of the star. The episodic accretion is probably powered by the circumstellar disk discussed here.