If you tried the window down/door handle test, and you did get an alarm, then the LED blinker has a problem. If you did not get an alarm, then you need to need to be pretty adept at electrical troubleshooting. You must use the service manual schematic and a multimeter to check this circuitry, and the troubleshooting guide in the service manual is helpful, too.
I would say that the switches on the various things that open are the likeliest culprits. For instance, you cannot arm the alarm if the trunk is open, likewise the hood. But, first check all 3 fuses called out in the guide: #6 and #8 in the main fuse box, and #45 in the main relay box. Then, follow the guide troubleshooting steps, and they should lead you to the problem.
Good luck,
Bill