|I'm assuming this is regarding a desktop computer.|
Why on earth do you think you need 3 optical drives ?
You can do everything with ONE combo DVD burner drive - reads and burns both CDs and DVDs.
Is the new DVD drive IDE or SATA or USB connected ?
It is common to un-intentionally damage IDE data cables, especially while removing them - the 80 wire ones are more likely to be damaged. What usually happens is the cable is ripped at either edge and the wires there are either damaged or severed, often right at a connector or under it's cable clamp there, where it's hard to see - if a wire is severed but it's ends are touching, the connection is intermittent, rather than being reliable.
Another common thing is for the data cable to be separated from the connector contacts a bit after you have removed a cable - there should be no gap between the data cable and the connector - if there is press the cable against the connector to eliminate the gap.
80 wire data cables are also easily damaged at either edge if the cable is sharply creased at a fold in the cable.
Try another data cable if in doubt.
80 wire data cables must have the proper end connector connected to the mboard IDE header - usually that's blue, but in any case it's the one farther from the middle connector on a 3 connector data cable.
Check your SATA data cables. The connector on each end should "latch" into the socket on the drive and on the mboard, or on the drive controller card - it should not move when you merely brush your hand against it near the socket - if it does, mere vibration can cause a poor connection of it - use another SATA data cable that does "latch", or tape the connector in place.
(There is a slight projection or bump on one side of the outside of the connector that "latches" it into the socket - it's easily broken off or damaged)
The same thing applies for the SATA power connection.