Maybe you don't know them by name, but you'd be surprised how many classical pieces you recognise.  The theme tune of The Apprentice is taken from Prokofiev's Romeo and Juliet (Prom 16, 27 July), while Prom 11 on 23 July showcases Wagner's cinematic Ride of the Valkyries, which was featured in Apocalypse Now, and at Prom 29 (6 August) you can hear Richard Strauss's Also sprach Zarathustra - used brilliantly by Stanley Kubrick in 2001: A Space Odyssey. 

In any case, a lack of familiarity with the repertoire need never be the obstacle to enjoyment of classical music in the age of YouTube.  If a piece of music is well known enough to have been played live, you can guarantee you'll be able to hear a version of it online.  Now is the time to enrich yourself.