Balmorra (Republic) Datacron Locations
Coordinates: X 192 , Y 4
This is by far the easiest Datacron for Republic players to find on Balmorra. Really all that you need to do for this Datacron is buy a key from a vendor for 5k Credits then open the Datacron chest. Anyway let's get started. The intro area that you land at on Balmorra (Bugtown) is actually where this Datacron is located. To start, far in the southeastern end you will find a vendor named Ba'teil (Screenshot 1 & 2). From her you will want to purchase the Lost Code Cylinder for 5k Credits.
Once you have the Cylinder you need to make it back to land. I've found the best way is to run north around the mountains and circle back around. This way puts you pretty much near the Datacron anyhow. With the Cylinder in hand you will want to go to the Space Chest in central Bugtown (Screenshots 3 & 4). Using the key on the chest will open it up and reveal the Datacron for you.
Coordinates: X - 1018 , Y 22
This Datacron is located inside the Okara Droid Factory in central Balmorra. If you forget where this factory is located refer to (Screenshots 1 & 2). Once inside you will be going to the western most elevator titled "Assembly Line". Take the elevator down and follow the hallways until you reach the giant room in the northeastern most end of the map. This is the Assembly Room, go figure. In here you will want to jump down to the floor below and make your way over to a super secret entrance in a column (Screenshots 3 & 4).
Inside this little room you need to drop down to the different pipes until you can see the Datacron. Once you can see the Datacron you need to fall down to the pipe that it is on (Screenshot 5).
Coordinates: X 729 , Y 173
This Datacron is located in Gorinth Canyon of Balmorra. To get here you will first need to go to Sundari Outpost and travel a bit south (go across the bridge into Gorinth Canyon) and make your way over to the hidden path that leads to this Datacron (Screenshots 1 & 2). This Datacron is so easy it isn't even funny... The only thing that could even pose a problem is the level 34 Champion nearby but if you're high enough level you can run right by him with 0 difficulty.
This Datacron is located in western Bugtown at the Colicoid Queen's Nest (Screenshots 1 & 2). Or, the coordinates are -163 , 6. Once you're here take the elevator down into the Colicoid Queen's Nest and make your way to the Queen's chamber. It is straight ahead from when you come down the elevator, in fact you can even see the Datacron the second you reach the ground floor. The secret passage to reach the Datacron can be found in the southern end of the Queen's chamber (Screenshots 3 & 4).
The secret passage way is filled with elite Colicoids to kill so be careful if you're lower level. At the very end you will see your Datacron reward sitting atop a fallen down pipe.
Coordinates: X -778 , Y 107
To reach this Datacron you first will need to fly to Lower Markaran Outpost in Balmorra. From here you will be following the road south through the Markaran Plains and across the bridge to Gorinth Canyon. Once in Gorinth Canyon you will notice an extremely large AA gun as well as a pipe that runs across the gorge (Screenshots 1 & 2). From here you will want to... You guessed it, go across the pipe to the large mountain-like thing and make your way across the hidden path.
At the end of the hidden path you will see the Datacron down below sitting harmlessly on a ledge (Screenshot 3). From here it is a simple task of just jumping down to the Datacron. Be careful though, if you miss a jump you will need to fly back to Lower Markaran Outpost and start this little trip all over again.