Yes, it will work. You will need to unlock the phone. If it's paid off, T-mobile will send you the unlock code and instructions on how to unlock it.
I have an unlocked T-Mobile Galaxy Note 3 and I am using a straighttalk AT&T sim. For the most part, it works fine with the AT&T sim but I am having a problem with it losing the data signal when traveling in a car, 4G data signal indicator disappears, gets really hot, and kills the battery. I still have a full phone signal when this happens. When it loses the data signal it will not reconnect to data until I restart the phone.. anyway, if you are happy with the T-mobile signal where you live, go ahead and buy the straighttalk T-mobile Sim and it should work just fine for you.