What kind of LED lights/wires are you trying to use?
If the LED's you bought work as both running lights and turn signals, and the ground is through the mounting stud - then you are wiring them wrong by connecting one wire to ground and one wire to power.
If the LED's you bought are single purpose and only have one bright light function - then most likely one wire is ground and one wire is power.
The stock turns signals lights do not use the mounting stud as a ground, and this may be causing you a problem with your new turn signals.
If the chassis wires are not burning up, then take your volt/ohm meter and test the wiring harness without any turn signals connected. You should have 12V of power available on the green wire for the right turn signal, and on the solid black wire on the left rear. (You will need to have the key turned on and the turn signal switch turned to the side you are testing). The ground wires are the black wires with white marks.
Link to Wiring Diagram.
http://suzukisavage.com/cgi-bin/YaBB.pl?num=1294779818