This is a common issue with most microprocessor development boards. Optimally, one would want the board to contain a ground pin for every I/O pin. But, with a limited number of GND pins, there is no great way to connect the grounds to the board, especially not with high frequency signals (> 1 MHz, perhaps). This is a very common issue for microprocessor dev boards.
If your connections are going from one board to another, you could get away with a single ground wire between the boards. However, if you actually need so many connections, you'll have to join them together somehow. As already suggested, soldering is an acceptable solution (where there is minimal vibration. vibration can lead to wires breaking.). You may want to look into terminal strips as your interconnect which would look more professional and be more reliable. Using (crimpable) butt splices, you can also crimp multiple wires together.
The best solution may be to make a custom PCB which does the connections for you (A protoboard could work, too). This would provide the best signal integrity.