Polymer optical fiber (POF) links typically have a limited bandwidth due to heavy modal dispersion. In order to achieve higher data throughput, advanced signal modulation formats and transceivers have been investigated for improved spectral efficiency. However, emerging new fibers such as microstructured POFs (mPOF) can further improve transmission performance by offering a larger bandwidth at lower loss. In this letter, we report multigigabit transmission over high-bandwidth mPOF. A gross data rate of 7.3 Gb/s is achieved after a transmission length of 50 m using discrete multitone modulation in an intensity-modulated direct-detection system using a directly modulated edge-emitting laser diode at 650 nm and an avalanche silicon photodiode. The achieved spectral efficiency is 3.4 bits/(s Hz) considering the uncoded net bit rate with bit-error rate below 10-3.