Brushes is where you can do this.
Drag your straight rectangular artwork (you will want a thin wide strip that will distort to a circle path) to the brushes palette swatches, and create a new art brush when asked. Then draw a circle, and apply your new art brush to the cirlce.
To create an art brush your artwork must be vector path. So if your artwork is bitmapped, you can use the live trace and then expand this to vector shapes.
Another method to try is to use an envelope distort of arc 100%. Thsi will give you a half circle, so you would have to create a top adn bottom half gfor your artwrok and stitch this together. The first method is better, btu have added thsi second mehtod incase you like working with envelope distorts.