Someone with more knowledge may correct me but I think that in the early days in the United States they were navigated using VOR/DME and/or ADF in conjunction with paper charts and dead reckoning.  Next came Inertial Navigation Systems (INS), then after the KAL 007 incident the federal government opened GPS signals for civilian use, and this was the impetus for modern day Inertial Reference Systems (IRS), GPS, and Flight Management Systems/Computers (FMS/FMC).  Of course, radio navigation aids and dead reckoning were and always are used as the underlying navigation method for safety purposes.  You can still navigate airways, SIDS, and STARS using VOR/DME, but your navigation chops, hand flying skills, knowledge of aircraft performance limitations, and chart reading must be well developed.