Controllability of linear continuous-time periodic systems is dealt with via a harmonic analysis approach for the first time in this paper. This approach reveals that controllability of continuous-time periodic systems can be connected to necessary and sufficient conditions expressed explicitly with Fourier coefficients of the system matrices, which can be interpreted in a way similar to what we have seen in linear time-invariant cases. These controllability conditions shed new light upon structural characteristics of continuous-time periodic systems that are hard to know by means of existing time-domain controllability criteria in the literature. Controllability canonical decomposition of linear continuous-time periodic systems is revisited through state coordinate transforms of strong analytic features. The results are heuristic and significant for examining structural characteristics of continuous-time periodic systems and extending controllability-related techniques that are widely employed in linear time-invariant systems to linear continuous-time periodic systems.