Everyone says it - "test early in the development cycle" - but very few do it. It is too late to do much when you have a container load of product. Test at the prototype stage.
Books have been written on the subject of circuit board design for EMC but I have a few hints that might make it easier to achieve good result- if you have high speed clocks ( > 20MHz.) use a 4 layer board, pay particular attention to decoupling capacitors on integrated circuits, be careful with switching regulator layout and where a wire has to enter or leave a board put a resistor or chip ferrite on the board close to the wire.
Wires including Ribbon Cables
To get an antenna with gain you need length. Rarely is there enough length on a board but there is plenty of length in the wires. Reduce the gain of the wire antenna by a ferrite on the PCB or around the wire. Do not forget the power cord.
Switching Power Supplies
This is a major cause of EMC problems- a few hints- put a resistor in the gate of the switching FET, 47 ohms is a starting point, use a 4 layer board if possible, keep the tracks with high speed signals on them short, put an RC snubber across any Schottky Diode. Pay particular attention to decoupling capacitors and where large value ceramics are used put a smaller value in parallel.