Structural properties and energetics of carbon rings are studied with the diffusion Monte Carlo (DMC) method. Our DMC-based geometry optimization reveals that both polyynic C-4n and cumulenic C-4n (+) (2) rings exhibit bond length alternations for n >= 3, which is understood to be due to Jahn-Teller distortions. The bond length alternation even in a cumulenic (4n + 2) carbon ring was experimentally observed in a recently synthesized C-18 molecule. From a comparison of the DMC cohesive energies of C-4n with those of C4n + 2, we present a comprehensive picture of the competition between Huckel's rule and Jahn-Teller distortion in small carbon rings; the former is more dominant than the latter for n < 5 where C4n + 2 rings are more stable than C-4n, while C-4n rings are as stable as C4n + 2 for n < 5 where dimerization effects due to Jahn-Teller distortion are more important.