Infrared Astronomical Satellite (IRAS) observations in 1983 revealed the existence of several solar system dust bands. These dust bands are believed to be debris produced by recent disruption events among main-belt asteroids, particularly because dust particles have short dynamical and collisional lifetimes. Using young asteroid families as tracers of recent disruptions in the main belt, we linked the most prominent IRAS dust bands with their sources. We propose that the source regions of the dust bands with inclination 9.35d and 2.1d are the Veritas asteroid family at 3.17 AU and the Karin cluster located inside the Koronis asteroid family at 2.865 AU, respectively. The Veritas family and the Karin cluster formed by collisional disruptions of their ~140 and ~25 km diameter parent bodies at 8.3+/-0.5 and 5.8+/-0.2 Myr ago, respectively. Asteroid material from the former source may represent about one-quarter of the interplanetary dust particles that have been collected in the Earth's stratosphere (and that have been extensively studied in laboratories). We were unable to identify a recent collision in the main-belt region that could be responsible for the 1.4d IRAS dust band. The region of the Themis family remains the best candidate for this dust band. We speculate that the (4652) Iannini cluster (<~5 Myr old, ~12° inclination) is the source for the J/K dust band and that the (1521) Seinajoki cluster (~15° inclination) is the ultimate source for the M/N dust band. We point out that the dust bands' spatial distributions are consistent with our proposed sources. This, and the fact that many prominent but ancient asteroid families have no associated dust bands, strongly suggests that dust bands are primarily by-products of recent asteroid breakup events that occur throughout the main belt.