As I recall, the classic cyclotron (Lawrence's original design) had the particle orbits enclosed in a pair of facing hollow "dees", with a gap between them, with the whole assembly between the magnet pole faces. An alternating potential difference between the dees gave the particles (which must have had to travel in bunches) a kick every time they passed through the gap.
The big insight that led to the cyclotron was the realization that the orbital period doesn't depend on the energy, in the non-relativistic regime, so a simple alternating voltage with a suitable frequency would accelerate the particles, at least until relativistic effects become significant.