Pretty much everything mentioned before is what we did. Our team duct taped the noodles on at several points using pretty thick layers of the stuff; the layers and tightness helped prevent the noodles from moving too extensively. Then we pulled the fabric on really tight to reduce further movement. Our woodboard was also cut to exactly two noodle widths (super precise metric measurement there) so there was no slippage of the foam.
This year our fabric was a dual-tone flip-velcro cover so that we wouldn't have to remove the bumpers to change our colour (life is SO easy when we need to change alliances now) but we then needed to ensure the noodles wouldn't slip around inside with the duct tape.
Now that I think about it, if your bumper has to be wider than the noodle widths, taping the noodles over and putting a line of adhesive (hot glue comes to mind) at the base would probably prevent the bottom and top from rolling around. Perhaps a strategy for next year...
Edit: We also taped the noodles to each other, helping improve their structural integrity.