To operate efficiently across a wide range of workloads with varying power requirements, a modern processor applies different current management mechanisms, which briefly throttle instruction execution while they adjust voltage and frequency to accommodate for power-hungry instructions (PHIs) in the instruction stream. Doing so 1) reduces the power consumption of non-PHI instructions in typical workloads and 2) optimizes system voltage regulators' cost and area for the common use case while limiting current consumption when executing PHIs. However, these mechanisms may compromise a system's confidentiality guarantees. In particular, we observe that multilevel side-effects of throttling mechanisms, due to PHI-related current management mechanisms, can be detected by two different software contexts (i.e., sender and receiver) running on 1) the same hardware thread, 2) co-located Simultaneous Multi-Threading (SMT) threads, and 3) different physical cores. Based on these new observations on current management mechanisms, we develop a new set of covert channels, IChannels, and demonstrate them in real modern Intel processors (which span more than 70% of the entire client and server processor market). Our analysis shows that IChannels provides more than 24x the channel capacity of state-of-the-art power management covert channels. We propose practical and effective mitigations to each covert channel in IChannels by leveraging the insights we gain through a rigorous characterization of real systems.