Answer: You have several options. One is to cut the tree roots where they enter the bed. This gives the plants "free run" of the flower bed, but must be repeated once or twice a year to keep it tree-root-free.
You can install a piece of galvanized metal flashing or very thick plastic root barier as a vertical, underground, wall around the bed. It should extend down about 18" for best results. Roots will hit the wall and then grow laterally or downward. Few will go under the "wall" and then back up to disturb plants in the bed.
Another option is to lay a plastic sheet over the soil and plant on top of it, as you suggested. This is alright as long as the soil is plenty deep above the plastic to allow good root depth of the plants in the bed. It is also important that excess water can drain laterally off the plastic rather than collect in low spots, causing waterlogged soil conditions.
A third options is to just build up the bed and plant in it. By the time the tree roots move up to the new soil your plants are well established. This however is obviously putting off the problem.
Good luck with your garden!
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