Why am I asked for my Keychain password when the keychain is already unlocked?
This happened to me after changing my login password. I fixed it thanks to article on Apple support: https://discussions.apple.com/thread/6724601

Simply I had to change password for my "login" keychain.

But I was asked to input password to open "Local items" keychain. I solved that issue exactly according this post: http://support.apple.com/en-us/HT203897
You need to open folder ~/Library/Keychains/, find and delete folder with name similar to "A8F5E7B8-CEC1-4479-A7DF-F23CB076C8B8" and restart your Mac immediately.

I had to fixed both "login" and "Local items" prompt to solve it. Only changing password for "login" keychain seemed not to work although it should according Apple support. But I haven't test it heavily. 

Hope it helps.