Elizabethan music comes from the reign of Elizabeth I of England in around the 16th century, when high culture truly took hold in England. It is a very rich genre; it all depends how far you want to get into it. You can dip one toe in the water by picking up something by John Renbourn, who swims in "folky" waters but combines Renaissance and Elizabethan influences with English folk ballads and blues (Big Bill Broonzy being a major influence). His latest, called Traveller's Prayer, is an excellent start; also consider his earlier solo works (Sir John Alot of..., The Lady and the Unicorn, The Hermit, Black Balloon), as well as his collaborations with Bert Jansch and Stefan Grossman, and his work as part of Pentangle.

You can also approach this music from a more "classical" perspective (McGrath, I agree that this category is a misnomer, but it's probably not worth battling against the tide), which gives you a better knowledge of the breadth of Elizabethan music. Try listening to the lute music and lute songs of John Dowland, the preeminent Elizabethan lute composer, as played by Paul O'Dette or Robin MacFarlane (or in early recordings by Julian Bream, who played it on both lute and guitar, solo and with a small consort). For larger ensembles, try recordings by the Waverly Consort or the Early Music Consort of London (with David Munrow at the helm). Most of this type of stuff is categorized by the record companies as "Early Music," along with medieval and Renaissance musics from countries other than England.

This may be more of an introduction than you're looking for, but if this ends up appealing to you as much as it does to me, you'll be pleased that there's so much good stuff out there. Hope all of our suggestions vis-a-vis the I-IV-V rut are helpful; we all hit ruts sometimes, so don't let it get you down. Good luck with everything.