1. A computer system comprising: hardware resources including a processor, memory, and a storage device that provisions a thinly-provisioned logical block device; and a hypervisor running on the processor and interfaced with the storage device, the hypervisor supporting execution of one or more virtual machines and emulating the hardware resources for the virtual machines including an emulated thinly-provisioned logical block device that is provisioned from the logical block device, wherein the hypervisor includes a component for issuing commands to the logical block device to deallocate storage blocks from the logical block device, and each of the virtual machines includes a component residing therein for issuing commands to the emulated logical block device to deallocate storage blocks from the emulated logical block device, and wherein the commands to the logical block device are issued based on a granularity value according to which the logical block device performs space reclamation operations, and the commands to the emulated logical block device are issued based on a granularity value according to which the emulated logical block device performs space reclamation operations.