The Salt River Landfill accepts Yard or Green Waste which includes leaves, tree branches, untreated and unpainted lumber, grass and plant clippings.  Regardless of whether your green waste is picked up curbside or you bring it to the landfill for disposal, it should NOT be bagged.  Yard and green waste is recycled and made into mulch or further processed and made into compost.

The City of Mesa offers green waste collection services to its residents, and provides a green-colored barrel for those who subscribe to the service.  Specific information can be found at:
http://www.mesaaz.gov/residents/solid-waste-trash-recycling-/trash-recycling-for-single-family-homes/green-yard-waste-program.

The City of Scottsdale provides monthly brush and bulk item collection.  Although the brush and bulk items are collected in the same truck, the City of Scottsdale has an agreement with the Landfill to sort the bulk items from the green waste, so the green waste does get recycled.  Specific information can be found at:
http://www.scottsdaleaz.gov/solid-waste/brush-bulk-collection

The Town of Gilbert collects green waste with their monthly Bulk Trash pick up.  Specific information can be found at: http://www.gilbertaz.gov/departments/public-works/environmental-services/bulk-trash