The biggest "mindset" difference between single-threaded and multi-threaded programming in my opinion is in testing/verification. In single-threaded programming, people will often bash out some half-thought-out code, run it, and if it seems to work, they'll call it good, and often get away with it using it in a production environment.
In multithreaded programming, on the other hand, the program's behavior is non-deterministic, because the exact combination of timing of which threads are running for which periods of time (relative to each other) will be different every time the program runs. So just running a multithreaded program a few times (or even a few million times) and saying "it didn't crash for me, ship it!" is entirely inadequate.
Instead, when doing a multithreaded program, you always should be trying to prove (at least to your own satisfaction) that not only does the program work, but that there is no way it could possibly not work. This is much harder, because instead of verifying a single code-path, you are effectively trying to verify a near-infinite number of possible code-paths.
The only realistic way to do that without having your brain explode is to keep things as bone-headedly simple as you can possibly make them. If you can avoid using multithreading totally, do that. If you must do multithreading, share as little data between threads as possible, and use proper multithreading primitives (e.g. mutexes, thread-safe message queues, wait conditions) and don't try to get away with half-measures (e.g. trying to synchronize access to a shared piece of data using only boolean flags will never work reliably, so don't try it)
What you want to avoid is the multithreading hell scenario: the multithreaded program that runs happily for weeks on end on your test machine, but crashes randomly, about once a year, at the customer's site. That kind of race-condition bug can be nearly impossible to reproduce, and the only way to avoid it is to design your code extremely carefully to guarantee it can't happen.
Threads are strong juju. Use them sparingly.