By combining data from the NASA Wide-field Infrared Survey Explorer (WISE) mission with optical spectroscopy from the W. M. Keck telescope, we discover a mid-IR color criterion that yields a 78% success rate in identifying rare, typically radio-quiet, 1.6 less than or similar to z less than or similar to 4.6 dusty Ly alpha emitters (LAEs). Of these, at least 37% have emission extended on scales of 30-100 kpc and are considered Ly alpha "blobs" (LABs). The objects have a surface density of only similar to 0.1 deg(-2), making them rare enough that they have been largely missed in deep, small area surveys. We measured spectroscopic redshifts for 92 of these galaxies, and find that the LAEs (LABs) have a median redshift of 2.3 (2.5). The WISE photometry coupled with data from Herschel (Herschel is an ESA space observatory with science instruments provided by European-led Principal Investigator consortia and with important participation from NASA) reveals that these galaxies are in the Hyper Luminous IR galaxy regime (L-IR greater than or similar to 10(13)-10(14) L-circle dot) and have warm colors. They are typically more luminous and warmer than other dusty, z similar to 2 populations such as submillimeter-selected galaxies and dust-obscured galaxies. These traits are commonly associated with the dust being illuminated by intense active galactic nucleus activity. We hypothesize that the combination of spatially extended Ly alpha, large amounts of warm IR-luminous dust, and rarity (implying a short-lived phase) can be explained if the galaxies are undergoing brief, intense "feedback" transforming them from an extreme dusty starburst/QSO into a mature galaxy.