Just a quick note of agreement on the Shostakovich quartets - not really as "modern" as what some are listening to here, but that's what makes them more "accessible". As Lousyreeds mentioned - the No.3 is a standout, IMO the Gabrieli Quartet on Decca or even London Treasury vinyl is spectacular both performance-wise and sonically.
Another recommendation in the modern, but not TOO modern, and still very tonal are Benjamin Britten's works, esp. Simple Symphony.
Another recommendation in the modern, but not TOO modern, and still very tonal are Benjamin Britten's works, esp. Simple Symphony.