I like thick layers of wet newspaper and then thick mulch but not piled up around the crown of the flowers. Replenish the mulch as needed, as it will break down, which is a good thing, and add organic matter to your soil.

A barrier of some kind, landscape fabric, paper (cardboard, newsprint, etc.), or plastic simply allows you to use less of the covering material. All of these block plants from access to the sunlight they need to grow so they may stop unwanted plant growth from the soil. However, there are some very persistent plants that nothing will stop.
Paper is biodegradable and often last only one growing season, if that long. Paper is most often readily available and free.
Landscape fabric costs, but does work in some instances. My experience is that after a few years persistent unwanted plants will punch their way through requiring removing the fabric, cleaning up, and replacing that material. Most landscape fabrics are made of nylon, or some other synthetic fabric material so they probably are not sustainable.
Plastic costs and is made of non renewable resources. Opaque Plastic will not only block sunlight but air and water from entering the soil. I have found soils that have been covered with plastic to be putrid smelling and lifeless.

No no no, do NOT waste the money on landscape fabric! I had my complete backyard dug up, the landscaper recommended using the fabric under the mulch, and the weeds still come thru the fabric. I'm gradually ripping it up and will use cardboard, a little more hardy than newspaper. My sis used old carpeting, but said the downside is you have to cut thru it when planting, but that it really did do away with the weeds.