I'm with JoG. Focus on the flasher wiring. See if there's power to the flasher relay. see if there's continuity between the can and indicator bulb terminals. check the switch has continuity. easy things but takes time. At least it's free. Use a test light or multimeter. You should buy a digital multimeter anyway. They're handy. You can make a tst light in minutes from a 12v bulb 2 bits of wire and 2 alligator clips. Or buy one.

The flasher relay is a small black box in a rubber holder located near the neck of the bike ,on the right side ,tucked under the forward part of the tank.It has two blades at right angles to each other,that plug into the connector ,I'd check here first as it controls both left and right signals ,you might also try to replace it with a auto relay for a test.(It'll plug right in)Do  you have stock turn signals ? do you have after market signals with duel filament front bulbs?More info would help!

Just checked the signal switch and getting power from both the hazards and regular switch.  When I remove the relay I no longer get power to the switch.

http://suzukisavage.com/cgi-bin/YaBB.pl?num=1294779818

Maybe not helpful to this post but here ya go anyway.

One fuse is for decompressor relay & selenoid and the other fuse is for everything else
Do the turn signals light up but don't flash or not work at all ?

ARE they hooked up right? there was three wires running to the front signals because they were double filament bulbs,the second filament used for running lights, the solid color wire on each side (gray I think) were used for the running lights, If your new signals are single filament these wires should be taped off and not used.The remaining wires should be two black and white grounds that connect to black wires coming from the new lights, and a yellow and brown wire that is connected to the red wires(guessing) that come from each light.The rears are the same, black to black and white ,red to yellow, red to brown.........JOG"s question of LED's is a concern because some LED's run off twelve volts while others run off lower voltage and need an adapter .LED's must be hooked up ground to ground, positive to positive ,if polarity is reversed, they will not function.

If you haven't another flasher unit yet,you could take a small piece of wire stripped at both ends , and insert it were the flasher goes,then try your signals ,they should light but won't flash,if they do your flasher unit is bad ,if they don't you have a wiring problem.

So I disconnected the back lights and the front onset started working.   But as soon as I connected either of the two back it shut off.  So I will tear apart the wiring to the back.  

If one wire is bad to the back, let's say the right blinker, shouldn't the left one work if the right is disconnected?  Or is there an issue further up the wiring harness?