After I was hacking with some files I get it.

Best method:-

1.To Remove the 'tile'

First of all, make a backup of the contents in /usr/share/unity/icons. To do so open a terminal by pressing Ctrl+Alt+t and type in the following:

mkdir ~/backup_icons
cp /usr/share/unity/icons/* ~/backup_icons

Then the files below, present in /usr/share/unity/icons are to be opened and edited so that all its contents are removed:

launcher_icon_back_54.svg
launcher_icon_back_150.svg 
launcher_icon_edge_54.svg
launcher_icon_edge_150.svg
launcher_icon_glow_62.svg
launcher_icon_glow_200.svg
launcher_icon_selected_back_54.svg
launcher_icon_selected_back_150.svg
launcher_icon_shadow_62.svg
launcher_icon_shadow_200.svg
launcher_icon_shine_54.svg
launcher_icon_shine_150.svg
round_glow_62x62.png
round_shine_54x54.png 

It might be difficult to open each file separately and edit. To avoid that do the following:-

• Open a terminal by pressing Ctrl+Alt+t . Then type in :-

  cd /usr/share/unity/icons
  

• Now type in :-

  for i in launcher_icon_back_54.svg launcher_icon_back_150.svg launcher_icon_edge_54.svg launcher_icon_edge_150.svg launcher_icon_glow_62.svg launcher_icon_glow_200.svg launcher_icon_selected_back_54.svg launcher_icon_selected_back_150.svg launcher_icon_shadow_62.svg launcher_icon_shadow_200.svg launcher_icon_shine_54.svg launcher_icon_shine_150.svg round_glow_62x62.png round_shine_54x54.png; do sudo inkscape $i && read -p

This should open each file one by one for you to edit. All you need to do is just select the image that is shown in Inkscape and delete it. Then Save it and close it. Now you need to press Enter(aka Return) in the terminal for the next file to open. Repeat the editing, saving,closing and pressing Enter process again and do the same till you have edited all the files. Now reboot your PC. You should have your launcher without the tiles.

NOTE:- I have used Inkscape to edit the files. But if you want to use Gimp(or any other editor) you can do so by just changing the word inkscape to gimp(or the name of any other editor) in the above command.

2.To add the 'tile' back again

For this you just have to copy the backup you created above back into /usr/share/unity/icons. For this open a terminal by pressing Ctrl+Alt+t and type the following:

cp ~/backup_icons/* /usr/share/unity/icons

You should have your launcher back with tiles now.

Alternate method:-

1.To Remove the Tiles

First you need to get to your ~/Downloads folder by the command:

cd ~/Downloads

Then download a file by the command:

wget https://www.dropbox.com/s/n48gz3g0fachi3t/icons.tar.gz

The download contains a icons.tar.gz file. Then make a backup of the image files at /usr/share/unity/icons by doing the following:

mkdir ~/backup_icons
cp /usr/share/unity/icons/* ~/backup_icons

Then get to /usr/share/unity:

cd /usr/share/unity

Now untar the tar file downloaded into /usr/share/unity by the following command:

sudo tar zxvf ~/Downloads/icons.tar.gz

This should get rid of the 'tile' .

2.To add the Tile back again

First you need to get to your ~/Downloads folder by the command:

cd ~/Downloads

Then download a file by the command:

wget https://www.dropbox.com/s/31ory2zycvk5c6x/icons.tar.gz

The download contains a icons.tar.gz file. Then make a backup of the image files at /usr/share/unity/icons by doing the following:

mkdir ~/backup_icons_notile
cp /usr/share/unity/icons/* ~/backup_icons_notile

Then get to /usr/share/unity:

cd /usr/share/unity

Now untar the tar file downloaded into /usr/share/unity by the following command:

sudo tar zxvf ~/Downloads/icons.tar.gz

This should add the 'tile' back.

PS:- Actually, the download file I have provided above contains the /usr/share/unity/icons folder with the above mentioned images edited(in the case of "Removing the tile") and unedited(in the case of "Adding the tile back again"). I had thought of saving you the work of editing the images and so I have provided the download files.

By the way, after removing the 'tile' the launcher is going to look like this.(Note: The trash icon will look funny because of my icon set.)

CREDITS for the "Best Method" goes to this question.

There is no easy way to do it, since it's hard coded. But a workaround would be editing the images that form the "tile" as you call it and change them for transparent (empty) images.

You can find them in:

/usr/share/unity/icons/

There you'll see a bunch of images for when the icon is normal, pressed, highlighted, etc. Edit them with gimp, for example, and replace the originals. Of course, make backups first.

I created a shell script on GitHub to do the task. Please check out the link. Tested on Ubuntu 15.04.

Another easy approach: Just install a custom theme.
Here you can find a few themes: http://www.noobslab.com/p/themes-icons.html
Some of them remove the "ugly by the 'square-tile-thing'" and look partly like the original Unity theme.

(You can switch themes for example with the Unity tweak tool)

The look and feel of the launcher icons are not directly related to Unity per se, but to the GTK+ theme you have currently selected for Unity, most likely the default theme.

Editing the default system theme is not a good idea since any changes you make are likely to be overwritten by an update. If you choose to go with this approach then consider cloning the current theme and giving it a different name.

Having said that there are already enough themes that don't have the launcher icons underlays.

I got satisfactory results with the mbuntu theme flavour from noobslab:

sudo add-apt-repository ppa:noobslab/themes
sudo apt-get update
sudo apt-get install mbuntu-y-ithemes-v5

## optional icons, which go well with the theme 
sudo apt-get install mbuntu-y-icons-v5

Next, install Ubuntu Tweak which is needed in order to switch the themes:

sudo add-apt-repository ppa:tualatrix/ppa
sudo apt-get update
sudo apt-get install ubuntu-tweak

Now launch Ubuntu Tweak, click Tweaks -> Theme, and select MBuntu-Y-Unity under Gtk theme and Window theme. Optionally select the Mbuntu icon theme.

Here is how the final result might look like: