Two problems with using the serger... first, you have much less control at getting your feed even, which is why your block edges don't line up. Second, you can't control the stitch length on most sergers and you need a very small stitch for quilt piecing. To add a third problem, the stitching is gathering up the fabric a bit and not letting it lie flat.
I'd use a sewing machine and be careful to have edges lined up exactly and an exact 1/4" seam. A walking foot would help you too, as has already been mentioned.