The CPU and all clocks will be disabled when LPM4 is set. See here.
Have you set this configuration in your .cfg file?
Power.allowDynamicMode = true;
From documentation for Power_changeIdleMode(), note that the CPU doesn't go immediately to sleep. The idleTask has to run for the system to enter LPM4 power mode. Is your application entering the idleTask (i.e. are your other tasks blocking. Only then can the system enter the idleTask)? You mentioned "My tasks are still running", how are you checking this?
Vikram