You need:
tt --> phono preamp --> headphone amp (e.g., LDMKIII) --> headphones
The phono preamp (phono stage) can be a stand alone unit or built-in to a preamplifier, integrated amplifier, receiver or even the turntable itself (e.g., one of the Pro-Ject Debut III models has a built-in phono preamp).
I use a vintage Kenwood preamp as my phono preamp and route one of the line out signals from a tape loop to my headphone amplifier. It works perfectly and sounds great. Better yet, I found the Kenwood on eBay for $50 and it can accommodate both moving magnet (more common) and moving coil cartridges. I also route my other sources (iPod, computer) through the Kenwood so that I can switch between sources without unplugging cables. It does a lot for $50. There are many older preamps that can perform these functions. As long as they have phono inputs that match your cartridge, you're all set.
Some headphones can act as 'preamps' (typically
not phono preamps) because they have a separate line
out signal that can be routed to a power amplifier and the signal is affected by the headphone amp's volume control.
In these cases:
source --> headphone amp --> headphones
and/or
source --> headphone amp (line
out with volume control) --> power amp --> loudspeakers
Looking at this picture:
http://i21.ebayimg.com/05/i/001/38/ef/c3f5_1_sbl.JPG
I believe this is the case with the LDMKIII. It does not have phono input capabilities on its own. You need a phono preamp between your tt and the line input of the LDMKIII.
You might also do some research on the these:
Phonopreamps.com Home Page
I haven't used one, but they have received favorable reviews on Head-Fi and elsewhere.
Separate phono preamps are also manufactured by NAD, Audio Technica, Pro-ject and Cambridge Audio - to name just a few.