There are two good answers here already, I'll add the relevant physics.
Any suspension has three key variables:
Its spring constant, i.e. how quickly the force rises as the suspension is compressed.
Its dampening, i.e. how much energy is lost in every compression/decompression cycle.
Its unsuspended mass, i.e. the amount of material that needs to move around due to the uneveness of the road.
Bicycles are a type of vehicle that requires very low energy losses (simply because typical energy input is limited to about 250 Watts). As such, bicycles absolutely need to keep the dampening low, because all the energy that you dampen away needs to come from your kinetic energy (conservation of energy!), which the rider worked hard for to produce. Now, the more unsuspended mass you have, the more dampening is required to keep continuous contact to the road. You don't want your wheels to bump up and down after riding through a pothole, that would be extremely dangerous. That's the reason why any suspension fork or car suspension includes shock absorbers: Their whole purpose is to drain energy, but they are required for safety.
Pneumatic tires have the great advantage, that only the rubber within the contact patch to the road is unsuspended mass. The entire rest of the tire, the rim, the spokes, the hub, the fork, they are all suspended, and thus do not require dampening. The tire itself also has enough dampening within its rubber to ensure a good contact. As such, you simply cannot find a suspension system that's more efficient than a pneumatic tire. In addition, pneumatic tires also allow easy adjustment of the spring constant by pumping them to different pressures, but that's just a bonus.
Now, alternatives have been tried and are even used by some. Most importantly by the use of airless tires. These replace the tube with a core of rubber foam. Unfortunately, the added rubber already increases the dampening significantly, and thus reduces the tires efficiency. Nevertheless, it's the next best thing to do after the pneumatic tire. All other alternatives that would move the suspension further away from the contact patch, and thereby massively increase the unsuspended mass, would lose too much energy to dampening, and leave the cyclist thoroughly frustrated.