In nonlinear microresonators with strong stimulated Raman scattering effect, it is difficult if not impossible to generate Kerr soliton microcombs with a small free spectral range (FSR) (< 100 GHz) due to the competition between the Raman and Kerr effects. In this article, we overcome this limitation by using odd-period photonic crystal microrings (PCMs). Numerical simulations on the silicon-on-insulator (SOI) PCM show that a small frequency shift (5 GHz) induced by the photonic crystal structure can moderately suppress the Raman effect, such that chaotic microcombs with a small FSR can be generated. With a larger frequency shift (e.g., >= 10 GHz), the Raman effect is significantly suppressed, and the soliton microcombs can be generated. For comparison, without the frequency shift, only Raman lasing can be achieved in a conventional microring. To investigate the applicability of the proposed method in other material platforms, we carried out simulations for the aluminium nitride (AlN) PCM. The results are comparable to those obtained on the SOI PCM. Our method opens a new approach to the generation of small FSR Kerr soliton microcombs in microresonators with strong Raman effect, which is important for expanding the available nonlinear platforms and applications such as telecommunications, radio-frequency photonics, and astronomical spectrographs.