I spent quite a lot of time investigating this last year. Last time I checked (around February 2012) the only realistic possibility of a traditional courier flight without knowing anyone on the inside or with an airline is with British Airways, who offer a courier spot on flights between London Heathrow and Tokyo Narita. I called up their reservations number, who put me through to the right department (BA World Cargo), who gave me prices and timing.
Whilst the flight price is about half what it would normally cost (between £300 and £450, though this does vary), there are certain conditions. You must return within 2 weeks (or you can extend to 3 weeks by paying an extra £50), and you need to book fairly far in advance - there's often a 6 month lead time or longer for these flights. Finally there is only 1 courier spot per plane - so travelling at the same time as a friend isn't possible. As far as I know Tokyo is the only destination available with BA as a courier, and BA is the only airline I could find that had a courier programme available to the general public
To be honest it's quite easy to get reasonable flights from London to Tokyo for around £600 anyway, without any of the stipulations or hoop jumping that courier flights entail. I think the dream days of jetsetting around the world for next to nothing via courier options on flag carriers are a thing of the past.
UPDATE: I'm just trying this again - the number to call is 0870 32 00 301 (BA World Cargo), however they are only open during standard business hours. I am going to attempt this again on Monday.
UPDATE 2: I just called - flights were available from April onwards (a mix of outbound dates), but as April is high season the cheapest flight available was just over £700. The salesperson told me that the lower tier of fares for April starts at £530, but none of these were available. This means the flight is about 2/3 of BA's normal fare. I just did a quick check on Kayak and, although indirect, Air France flies flies LON > CDG > HAN for £524 on the days I investigated. So basically no great savings (aside from that its direct).