According to Greg Kuperberg

it’s quite striking that real computers are very close to two-dimensional, and yet they are mostly used in a RAM machine mode with an emulation of complete circuit connectivity.

and here

On the other hand, transistors in real computers are not very far away from melting. Even though many computers look 3-dimensional, most of the geometry of a computer is within each chip of the computer, and that geometry is almost completely 2-dimensional. One reason for that is the photolithography used to make the chips. But another reason is that there is no way to carry away the heat from a 3-dimensional block of transistors. Without that problem you could sandwich many chips together in a sort-of 3-dimensional pile. The heat problem effectively limits real computers to the power of 2-dimensional cellular automata. However, this 2-dimensional geometry is mostly used to simulate a RAM machine. It cannot be an efficient simulation, but it is what happens in practice, since most higher-level languages create a RAM machine environment for software. It’s also a pain to design algorithms for a 2D computational grid rather than for a RAM machine.

According to Joe Fitzsimons

The rate at which a region of space can be cooled scales as the surface area, where as the heat produced scales as the number of irreversible gates. For a 2D array these scale in the same way, but for a 3D array the heating scales as the volume (R^3) where as the cooling scales as the surface area of a bounding box (R^2). Clearly you need to balance the rate at which heat is produced with the rate at which it is removed, and hence you have a scaling problem with 3D arrays. This is entirely independent of the cooling mechanism.