The phrase "unimportant part" is correct and usable in written English.
You could use it when you are speaking about something that isn't as significant as other areas. For example: "The unimportant part of the project was finished quickly."
For most firms offices were a necessary but relatively unimportant part of their infrastructure.
"And I was a very small, unimportant part of it, but I was a part of it, and that implies responsibility".
During the colonial period the Yucatán remained a remote and unimportant part of the Viceroyalty of New Spain (centred in Mexico).
Under the Kalmar Union, Norway became an increasingly unimportant part of Scandinavia politically, and it remained in a union with Denmark until 1814.
Or, as Kristol said, "I think Bush's speech will be a not unimportant part of the dynamic of the election, with the caveat that it's always hard to predict what's going to be important".
Go to the Divan Parisien and ask for John (the Matchless) and tell him you want some of his superb chicken Mornay Florentine in which the broccoli is made to play a secondary but by no means unimportant part.
Meanwhile, his deep budget deficit leaves little cash to spare for sorting out strategically unimportant parts of the world.
At an appeal for one of them, prosecutors argued that this did not matter, because the man's attorney only missed unimportant parts of the testimony.
A network size reduction methodology is introduced to quicken the simulation time by eliminating the unimportant parts of the network.
In other words, it is a way to keep the significant information of the input data whilst getting rid of the unimportant parts.
"What I find interesting about these is the distortion as a result of selective emphasis; parts of interest are emphasized, unimportant parts reduced or left out.
