I have written elsewhere here that I do not think that having a "good" voice is any more of a prerequisite for becoming a good singer, than having a good guitar is for becoming a good guitarist. I agree with the earlier comments that we tend to start off as imitators. It really does take some time for most of us to get used to the sound of our own singing voice. I think that when we learn to start liking that, we immediately become better singers.
We start off by listening to other singers and then we hope we sound similar. As Jerry hints at, what really makes us better, is learning to sound like ourselves.
The other comment I liked here was the one about letting the song do the work. I remember some twenty years ago, when my mother played me a CD of the opera soprano, Kiri Te Kanawa singing jazz standards. I usually hate "posh" voices trying to "slum it". It was actually quite enjoyable though. The woman had the good sense to show the song rather than to show off her voice.
I don't really know where my "own" sound comes from. I try to make it sound believeable enough that I am the character in the song, which I am singing at the time. Getting the song across is the priority, not the sound of my voice. I think I try to get some of the timbre of the folk club singers, whom I heard in the seventies. I don't think my voice sounds like that though.