I've gone through this in my garden after a brutal summer and honestly, a fine-toothed rake works well, but what people forget is that you have to do it when the soil is moist - if you do it with completely dry dirt it becomes a mess because the dead grass breaks into a thousand pieces instead of coming out cleanly! another detail: don't press too hard, the goal is just to lift that layer of dead grass without scratching the roots of the living one. if you have large areas that are really affected, you can also consider a scarifier (that motorized machine people mention) but really, for small or medium-sized patches a good rake does the job without complications and you can see better what you're doing.