Over the winter, I use crimson clover (not to be confused with red clover - which will be a bad idea to plant in your garden) and sometimes fava beans. I usually will turn in the whole plants by hand in the spring a few weeks before planting, giving them time to break down in the soil. For warm weather, nothing beats buckwheat.
It's important for soil health to always be growing something (bare ground is no good), so if you're not going to be growing anything in a plot for a while, plant a cover crop. During the growing season, buckwheat can go from seed to turn-in in less than a month, making a bed ready to plant again in potentially 5 weeks.
Cover crops will suppress weeds, break up clayey soil, fix nitrogen, and raise the organic matter content of your soil. But don't be fooled, the nitrogen fixed is not enough to replace proper fertilizer. You still need to fertilize. There's no reason not to cover crop in my opinion, but you do need to be careful what you cover crop with and be aware of its behavior.
For example, if you let buckwheat flower and start to set seed, the stems will turn woody and won't rot very easily, making your bed unplantable (or you'd have to just mow it and remove the green matter, letting the roots rot, which would be fine but it's nice to get the green matter in your soil to raise the organic matter content). Generally same thing goes for crimson clover or favas, although with favas you can let them go to seed and produce a crop if you like.
I live in a mild climate where I over-winter a lot of veggies, and I'll toss crimson clover seeds in those beds with the veggies (kale, spinach, broccoli, lettuce, carrots) and let it sprout in the fall. The clover will be non-competitive till spring when the veggies are just about done, the clover takes over, and then I turn the whole thing into the soil and let it rot.
If you live in a colder climate with hard-freezing winters you might need a different strategy. Here in the pacific northwest, if you left your garden bed bare all winter while the rains pounded on it, the soil would be leeched of nutrients and totally compacted in the spring. A cover-cropped bed is soft and healthy.