Here is a neat trick for those of you who might not know (I just learned this today).

To remove triangulation from your model, go to Model > Modify > Poly Mesh > Quadrangulate

There might be some triangles left over but for the most part your model is easy to work with again.

Depending on the model some tweaking of the quadrangulate parameters can get you fairly close to your desired output. Thanks for the pro-tip!

Well. I know this question may makes some people like "Oh. Really? (facepalmed)"-But what does the triangulate do? Is it important and what does it do?

Triangulation turns all polygons into triangles. Quadrangulation turns the triangles into quadrilaterals, making them easier to work with.