You don't want to put plastic under your soil if it's in your flower
beds - it keeps light, air and water out and that's not good for your flowers
I never thought about using burlap
but that sounds like a good option since it will let the water and air through. They sell 'landscape fabric' made just for flower
beds that you can use too
Is composting an option for you? You could put down cardboard and then build a lasagna bed to smother the grass/weeds
and have a great bed next spring
Then just cover with mulch, or even use compost to dress your beds. It will keep your soil crumbly and easy to pull any weeds
that sneak in.
Hope that helps!