To complete the other answers in case other people have this problem: I also had this problem, all those solutions did not work.
I just had to... change the USB socket in which I plugged my cable.
Actually, it took some time to find that because Windows was correctly recognizing the device. But not ADB.
For Samsung Note 3 specifically, you will need to switch to "Media Device USB3.0" instead of just "Media Device MTP" before the device show up on adb devices.
I had same problems and tried everything, and that worked for me, Sometimes the problem is in the drivers; try this :
http://adbdriver.com
download the adb driver and run it, it will download the driver for your phone,
There're many reasons which can cause the problem, try:
Mine finally recognized the device after changed USB connection mode to "Camera (PTP)" Settings > Storage > (upper right hand corner more options) USB computer connection.
I don't know if it's because I rooted mine.
Might be faster just to restart the adb server:
adb kill-server ; adb start-server
Or what I often use:
adb kill-server ; adb devices
When you call adb devices
it automatically starts up the adb server. Then gives you the list of devices it can find. Sometimes even then it doesn't find the device right away so you may need to call adb devices
a few more times.