Besides fat low pressure tires, which don't fit all road bikes or indeed most road bikes, especially up front, there are a couple other things:

Carbon frame and fork. Not the biggest bang for the buck, but quite effective. There's a lot of variation between brands, worth testing a few on rough roads with same or similar tires and pressures. That last is important.

Light hands, a good bit of saddle to bar drop, and plenty of reach. This fix is almost free and makes the bike a whole lot more fun.
You want your saddle back far enough that you can be riding down the street on the hoods and briefly lift your hands off the bars without sliding forward on the saddle.
You want your bars at or below saddle height, measured with a level. 10cm is a nice amount of drop. That gives you an nice hinge at the hips which takes the sting out of big hits and lessens perceptible vibration at the head and shoulders.
You want enough reach that your upper arms make a ~90° angle with your straight torso. That's while looking in a mirror with your hands on the hoods and your forearms level. That, when added to the already light pressure on your hands, allows your hands to easily move up and down. One does still notice vibration, but it doesn't bother than hands or arms anymore.

The above is the traditional road fit which has been in use for over 100 years. It's not just because of wind resistance. It's also the most comfortable, which is the big reason for its almost universal adoption by recreational road riders.