All the info I gathered from various community sites and my own exploration (see "sources" page) is collected in this Google Spreadsheets document I have shared:
https://docs.google.com/spreadsheets/d/1-HtqVsrTVbk-Z6wUTJhG...I'll be keeping it up to date as new discoveries crop up.. if you just care about the achievement you only need to look at the first page though. The other pages provide details (spoliers) about the story behind the listening post transmissions.
The posts are listed in the order that I visted them in an effort to minimise travel time ... if like me you intend to do the loop visiting all 48 listening posts it'll take you on a
4676.44 Ly trip around the edge of the bubble. In the systems themselves you'll find yourself flying another
32768.2 Ls to reach the listening posts themselves.
As there are still only 16 discovered listening post quests/stories you will need to revisit two of the posts a second time if you want the achievement right now ... for me, allowing 18 days to pass before re-scanning the first two posts I visited was enough to pop the achievement.The number of jumps will vary depending on the ship you have, any half-decent jump range ship will suffice as you are not travelling into deep space, I wouldn't attempt this with less than a 25Ly jump range personally.
If you are doing them all in one hit as I did then you really must take a decent fuel scoop to eliminate downtime. You'll be able to keep your tank filled up easily as long as you filter your route to scoopeable stars only (FOG KBAM), you can easily complete the route using only scoopable stars. Add all the systems as bookmarks before you start off and then simply follow the same route I took.
When a listening post is not within 1000Ls of the main star in the system it will not show up in the nav panel, you should be able to use the information in the spreadsheet to plot your way to the correct planetary body easily enough.
The screenshot below will give you a rough idea of the route and what your system map should look like after you have set up your bookmarks:
http://imgur.com/sXtBfXFFWIW, I took my Asp with a 6A fuel scoop and ~45 Ly jump range and it took around 5-6 hours to visit all the systems over a couple of sittings playing in the background while I did other stuff.