In order to accomplish this, the process is broken down into three steps:
  - Turn off the original icon
  - Set the position, size & universal settings for the replacement icons
  - Insert the individual replacement icons

The CSS:

/*TURN OFF ORIGINAL ICON*/
#posts_control .posts,
#dashboard_controls_open_blog .followers,
#dashboard_controls_open_blog .drafts {
  background-image: none !important;
}

/*SET POSITION, SIZE & UNIVERSAL*/
#posts_control > .posts:before,
#dashboard_controls_open_blog .followers:before,
#dashboard_controls_open_blog .drafts:before {
  position: absolute !important;
  content: "" !important;
  top: 2px !important;
  left: 4px !important;
  width: 34px !important;
  height: 34px !important;
  background-size: 34px 34px !important;
  border-radius: 4px !important;
}

/*INSERT REPLACEMENT ICONS*/
#posts_control > .posts:before {
  background-image: url("http://25.media.tumblr.com/tumblr_mat0r0P5e21qlzdwqo1_100.png") !important;
}
#dashboard_controls_open_blog .followers:before {
  background-image: url("http://24.media.tumblr.com/tumblr_mat0r0P5e21qlzdwqo2_100.png") !important;
}
#dashboard_controls_open_blog .drafts:before {
  background-image: url("http://24.media.tumblr.com/tumblr_mat0r0P5e21qlzdwqo3_100.png") !important;
}