@Cam, @Willem, @Amstor and the rest of the tribe, You are ALL AWESOME!
Thanks a bunch for all the suggestions you made, as that shows a very high standard on the consideration arena.
So, replying to your questions/suggestions:
I have used Chrome and Firefox, with no luck.
My cookies are enabled on both Browsers, that's for sure, and nothing works with 3.7.1.
I have an IPAD (with cookies ENABLED) which has the same problem with the 3.7.1 version.
I tried to download version 3.7, get the wp-login.php and apply it to my 3.7.1 installation, with no luck again.
Yesterday I have been trying to have this problem solved, and it took me around 3 hours to just give up and go to bed.
Today I decided to start from scratch, and so, I completely erased my installation, and installed version 3.7, ET VOILA! It worked like a charm!
So, my dearest friends, 3.7.1 has definitely a problem, and I will not take anymore time to understand what is it.
I WILL NOT upgrade to 3.7.1, and will only upgrade to the next version, AFTER reading the comments on forums like this and be assured I will not get stuck.
Wish you all the best of lucks and if I discover something about this I will post it here, as a comment.
Love you all!