dahlia1972 @blown_dry has great suggestions here! When I started raised beds in my vegetable garden I did the cardboard thing too and covered that with a mix of approximately 30% compost and 70% screened topsoil. If you go the cardboard route, make sure to peel off any labels or packing tape, as those will stick around and annoy you later. Cardboard is actually pretty awesome not only for killing the grass, but also the glue that is used in the cardboard pressing process attracts a lot of earthworms which do their thing and improve your soil. Now my veg raised beds are only 4" tall, so it wasn't too expensive to fill them with ordered soil and compost. If you have really tall ones, her suggestion of filling the bottom portion with branches is a good one because it's healthy for the soil plus it takes up some of the space. If you want to go a step further, read up on the hugelkultur method! 😊
My husband designed and built my lovely vegetable garden fence and gate with a little help from me. His health isn't so great these days so I'm very thankful and grateful for that gift. Anyway, you can see the raised beds here. 