It looks like flight aware is using Mercator... which is what I assume most navigational charts use. If it is Mercator, then straight lines on the map are lines of constant bearing in reality, so pretty straight forward to follow.
I think the premise was that the actual route flown might have been much more complicated than it looks, if the map projection used by the aircraft wasn't the same as that used by the tracking service.
People in charge of the database is able to load arcs with a given radious. It's not possible to do it from the mcdu (the flight computer keyboard and display). I guess this was designed on the ground and then loaded at the aircraft.
since the flightplan shows a certain number of lat/long style waypoints you can assume they did this. Entering arcs in the FMS is somewhat possible (by approximation, i.e. multiple FIX<radial>/<distance> points); but entering these digits that way would be unpracticle (albeit NOT impossible).
Initially I looked at the site from the Iphone so I didn´t see the details. In fact is as you say, just waypoints set in a square way. The rounded corners is just the normal turning radio of the B747.
In theory you're right; but on the scale they flew this pattern; maintaining a constant heading will look like a straight line. Also, as mentioned in other replies; the map projection has a big impact on this (e.g. A lambert projection will show a constant heading as a curve).
Isn't it hard to constantly fly a slightly curved route?