I don't think you're going to find any magic workaround for this... if you want the pieces to be separate then you'll have to make separate pieces (at least in maya). You may be able to get away with a single texture file by grouping all the pieces you create then UV mapping to the group instead of each piece individually. Even if mapping individually, you can reuse the same texture by adjusting the UVs to only use the portion of the texture they need.

You might consider using Illustrator to trace the paths of each piece then import that into Maya as a curve that you can extrude and cap. I think there is a way to bring photoshop path into Maya also, I've never done it personally.

Good luck and let us see how you ultimately go about this.