If I look at the link above to the schematic that was posted and you are using separate 12vdc supply then yes...
Vcc on relay board to 5vdc Arduino, when the INO inputs on the relay board are grounded to Arduino GNd (or via your digital output pins) then this should "switch on" the relay as the opto isolated "trigger" circuit via the opco coupler is completed/activated and the ino led on the relay board is lit.
Do not connect GNd on relay board to GNd on Arduino but connect GNd on relay board to gnd on separate 12v power supply and make sure your 12 Vcc supply to relay board is connected to the right side of JD-VCC.
If you do it this way you will NOT be connecting either Vcc or GNd of either your circuits together which should be safe.
Overall, provided the schematic I saw is what you are using is correct then it seems your connections are o.k.
If you now touch each ino relay board input in succession to Arduino GNd each relay in succession should activate.
If this is the case then it is up to your code to do the switching to the right Arduino pins you chose to use.
..... If you are having more difficulty......
You mentioned you tested the relays manually and they worked so I am not sure if you may have possibly shorted something out, as it can't be ruled out I suggest the tests below.
Just do this test on each channel to see if anything is blown.
If you touch ino pins to the ground of Arduino, does the relays activate or just the ino led come on?
What is the voltage between pin 4 & 3 of the opto coupler when ino led is on?
What is the voltage between the collector and emitter of the relay switching transistor if the voltage between pins 3 & 4 of the opto coupler is 0?
If the voltage drop between pins 3 & 4 of the opto coupler are the same as the external voltage supply when the ino led is on it means the base emitter of the relay switching transistor is not forward biased and you have a opto coupler fault, if the pins 3 & 4 of the opto coupler are 0 or close to 0 volts when the ino led is on and the voltage between the collector and emitter of the relay switching transistor is same or close to external power supply it means you have a switching circuit fault.
Move to and test next channel.
The D1 diode is just for back emf protection when the relay switches off, but the diode should not be shorted, de solder one pin to check, if the diode is shorted your relay will not work.
Also check value R2, if open circuit the base emitter of the relay switching transistor q1 will not be forward biased, check value of R2 within tolerance.
Typically transistors, diodes and opto couplers are more suspect in fault conditions, either open circuit or short circuit, check for things that may get more than normal operating temperature.
..... tip.....
If you want to "invert" the on state of your relays to when the digital pins on the Arduino are on or "high", which is a better safety condition or "safe fail state", then use of 2 resistors and a transistor to ground Opto ino would be better, just search my name for my posts and you will find the circuit I Posted.
Or, located on the URL below.
http://www.ipcamtalk.com/showthread.php/4201-Arduino-amp-Blue-Iris-Integration