If you follow the FAQ link from JasonAzze, you will see there is a "map to guest" line which is also required, so you need both of these lines:

security = user
map to guest = Bad Password

I had the same problem as the OP, and I have tested that this solution works on Fedora 18

The security = share mode was deprecated since Samba 3.6, and has been removed from Samba code in February 2012 (commit d7bb961859a350):

s3-auth: Remove security=share (depricated since 3.6).

This patch removes security=share, which Samba implemented by matching
the per-share password provided by the client in the Tree Connect with
a selection of usernames supplied by the client, the smb.conf or
guessed from the environment.

The rationale for the removal is that for the bulk of security=share
users, we just we need a very simple way to run a 'trust the network'
Samba server, where users mark shares as guest ok.  This is still
supported, and the smb.conf options are documented at
https://wiki.samba.org/index.php/Public_Samba_Server

At the same time, this closes the door on one of the most arcane areas
of Samba authentication.

OK, I have found an answer myself.

As this is absolutely not obvious from the docs and HOWTOs and whatever, the reason this thing asks for password is because it cannot map guest user to the owner of the directory being shared.

I have NTFS partitions which I need to mount RW so I used the following setup in my /etc/fstab:

/dev/sdb1  /media/disk1  ntfs defaults,noexec,noatime,relatime,utf8,uid=1000,gid=1000 0       2
/dev/sdb2  /media/disk2  ntfs defaults,noexec,noatime,relatime,utf8,uid=1000,gid=1000 0       2

The most important pieces of config are uid and gid (maybe only uid, don't know). They are set to the UID and GID of the user jonnie set up on the server (obviously not root). So, when ntfs-3g will mount these disks, everything will be owned by him.

After that, I have added this user to the Samba registry (or maybe created new identical one, don't care):

# smbpasswd -a jonnie

It asked for password, I have entered the same as for the main system.

After that, I have added the force user and force group settings to the smb.conf:

[global]
  workgroup = WORKGROUP
  netbios name = HOMESERV
  security = user
  map to guest = Bad User

[disk1]
  comment = Disk 1 on 400GB HDD
  path = /media/disk1
  browsable = yes
  guest ok = yes
  read only = no
  create mask = 666
  directory mask = 777
  force user = jonnie
  force group = jonnie

[disk2]
  comment = Disk 2 on 400GB HDD
  path = /media/disk2
  browsable = yes
  guest ok = yes
  read only = no
  create mask = 666
  directory mask = 777
  force user = jonnie
  force group = jonnie

So, most important piece of config relevant to me was force user.

Courtesy of the Samba HOWTO