Does it use some kind of CPU interrupt
Yes! Modern preemptive kernels are absolutely dependent upon interrupts from hardware to deliver good I/O performance. Keyboard, mouse, disk, NIC, USB, etc. drivers are all entered from interrupts and can make threads that are waiting on them ready/running when required (e.g., when data is available).
Threads can also change state as a result of making an OS call that changes the caller's own state of that of another thread.
The interrupt from the hardware timer is one of many interrupt sources and is only special in that many system operations have timeouts that are signaled by this interrupt. Other than that, the timer interrupt just causes a reschedule which, in most cases, changes nothing re. the ready/running state of threads. If the machine is grossly CPU-overloaded to the point where there are more ready threads than there are cores, there is a side-effect of the timer interrupt that causes CPU time to be shared amongst the ready threads.
Do not fixate on the timer interrupt—the other driver interrupts are absolutely essential. It is not impossible to build a functional preemptive multithreaded kernel with no timer interrupt at all.