14 CFR 61.65(d)(2) says:
(ii) Instrument flight training on cross country flight procedures,
including one cross country flight in an airplane with an authorized
instructor, that is performed under instrument flight rules, when a
flight plan has been filed with an air traffic control facility, and
(A) A flight of 250 nautical miles along airways or by directed
routing from an air traffic control facility;
(B) An instrument approach at each airport; and
(C) Three different kinds of approaches with the use of navigation
The wording is saying that you must follow airways or a route given by ATC. But since the flight must be under IFR, you'll get a route from ATC in your clearance anyway so I don't know why the FAA has the airways reference in there.
(As a complete guess it may be left over from a time when there was less radar coverage and the FAA wanted to make sure that instructors weren't taking any shortcuts.)
I don't know what flight following has to do with this because that's VFR, and the flight must be IFR.