Weakly-coupled mode division multiplexing (MDM) technique is a promising candidate for capacity enhancement of short-reach optical interconnections, for which the multiple-ring-core few-mode fiber (MRC-FMF) has been proven to be an effective design method to suppress distributed modal crosstalk. Similar to low chromatic-dispersion (CD) O-band transmission based on single-mode fibers (SMF), all the mode channels in a weakly-coupled FMF for short-reach applications should achieve low CD to support intensity-modulation/direct-detection (IM/DD) transmission. In this paper, we propose, for the first time to the best of our knowledge, an index perturbation method to adjust both effective index and CD of each mode in an MRC-FMF. Firstly, an accurate modeling method to model the relationship between SiO2-GeO2 material index and the germanium concentration at different wavelengths is proposed by analyzing the index characteristics of 4 kinds of germanium-doped fused silica SMFs at the same fabrication processing, which could be utilized to calculate the CD characteristics for an MRC-FMF with perturbed index profile. Then, based on the perturbation method considering the influences on both effective index and CD, a weakly-coupled low-CD MRC-FMF supporting 4 linearly-polarized (LP) modes is designed and fabricated. The measured minimum effective index difference min|Δneff| among all modes is larger than 1.3 × 10-3, and the CD values of all the modes lie between -6 and +6 ps/km/nm ranging from 1280 to 1320 nm, which agree well with the design. The 2-km transmission experiment indicates that the fabricated MRC-FMF could support stable digital-signal-processing (DSP)-free IM/DD transmission for all the 4 LP modes. This work is beneficial to the application of short-reach weakly-coupled MDM systems.