Here are a few things that you might try:
1. Make sure you are using a power supply rated for sufficient voltage (5V) and current (1A according to the User Manual, but I use 4A-5A) for the RIoTboard.
2. Even if the power supply is rated correctly, use a voltmeter to test the output voltage of the power supply to make sure it is not faulty.
3. Use a serial-to-USB cable connected to the J18 Serial Debug Port to view the console output during boot. If there is an issue with USB (or something else) on the board, you may see something in the output that can give a clue as to the problem. While the UART8000-U cable is the recommended serial cable for the board, you can probably use just about any serial cable provided it is intended for use at 3.3V levels. Here are some references for using a serial-to-USB cable on RIoTboard:
Good luck on the troubleshooting!