If you're comfortable using it, go for it. It also depends on the country you are visiting. If you think you won't have trouble there using your national passport (for example, maybe as an asylee from Russia, you might be wary of using your passport in Belarus because they are close and might share information). Remember, they might also ask you at your naturalisation interview why you used your passport. This does not always happen, but it might. You'll have to tell them something like, you asked CIS, and they said it was ok since you were already a permanent resident, and not an asylee anymore, and that you did not feel afraid of using the passport in other countries.
If you are afraid to use it, then apply for a Refugee Travel Document (RTD) or a Re-Entry Permit (REP).