I have been growing in straw bales for a couple of decades, it is the only way to garden if you ask me. Stay away from bales of hay if you can, hay has seeds and those seeds sprout when watered, while straw should be virtually sterile because the seeds are harvested prior to baling the stalks. The raised height makes it much easier to plant and harvest. They heat up in the spring as they begin to decompose, then stay warm for 4-6 weeks in the early part of the season after I plant or seed into them. I get no weeds. They hold water really well and are a breeze to water with a soaker hose running down each bale and a battery hose end timer. You ask if the crops grow as well in straw as they do in soil, well the actual fact is that you are not growing in straw, since the straw is decomposing quickly and becoming "soil" so the answer is that everything does as well or better in the straw bales. No need to rotate crops, since you'll usually start with new bales each spring, and that means you are creating new soil inside each bale every year, so no lingering insect or disease issues that can accumulate in soil. Also, the compost you end up with the spring after you have used up the bales is incredible. I use it to fill my pots, window boxes, and containers, and to enhance the soils around my annual beds and perennial gardens. It is better than any bagged potting mix, weed free, and costs nothing!
Try it, you'll never go back to traditional gardening in soil.