When doing rounded corners you really need to use strips of fabric cut on the bias. This is because, as you sew around the corner, the outer fabric needs to stretch more than your sewn line.
Most quilters use straight-of-grain fabric strips to bind quilts as it handles much more easily than bias-cut strips. For quilts the corners are typically mitered, so there is no need for the stretch. Quilts with scalloped edges require bias fabric strips for the stretch.
You can bind with either straight-of-grain fabric strips or bias-cut fabric strips; however, they handle differently.