The short answer is that you can’t. When you write assembly code it runs sequentially (or with branches) on one and only one logical (i.e. hardware) thread. If you want some of the code to execute on another logical thread (whether on the same core, on a different core on the same CPU or even on a different CPU), you need to have the OS set up the other thread’s instruction pointer (CS:EIP) to point to the code you want to run. This implies using system calls to get the OS to do what you want.
User threads won’t give you the threading support that you want, because they all run on the same hardware thread.
Edit: Incorporating Ira Baxter’s answer with Parlanse. If you ensure that your program has a thread running in each logical thread to begin with, then you can build your own scheduler without relying on the OS. Either way, you need a scheduler to handle hopping from one thread to another. Between calls to the scheduler, there are no special assembly instructions to handle multi-threading. The scheduler itself can’t rely on any special assembly, but rather on conventions between parts of the scheduler in each thread.
Either way, whether or not you use the OS, you still have to rely on some scheduler to handle cross-thread execution.