The only time I trim fabric threads is if it is a dark thread under a white fabric. As Ginger mentioned, probably no one else would notice in the finished product. Threads that don't show through can stay just as they are. Often colored threads don't show enough for me to bother with; it is only really black or very dark threads that I will nip off, and then only as a last step before sandwiching. It does help to use good quality fabric, and it does help to starch.
One fabric I don't want to use again is Kaufman Kona solid black, as it creates way too many strings when cut into strips for sashing. Fortunately, the quilt I used it on was made up of very colorful batiks so I didn't have to worry about black showing through a solid white. Even the light batiks were dark enough for the black to not show through enough to worry about.