On his sixth studio album Momentum, though, he's moved closer to pop, and only featured two covers.
There's a definite '60s vibe to many of the tracks - opener The Same Things, and Everything You Didn't Do both feature the sort of electronic organ and drumming popular in that decade.
But there is a range of genres here - Anyway is RnB-tinged, Sad, Sad World is reminiscent of piano-based bands like Coldplay and Keane, while his cover of Cole Porter's Love for Sale is a deconstructed funk version, with a guest appearance by British rapper Roots Manuva.
Cullum's strength is obviously his talent for keyboard instruments, but he also has a way with words, writing lyrics like "I don't need a vision, I'm just waiting on collisions of the brain and the heart".
Fans of Cullum's jazz may be reluctant to give this more poppy album a try, but do: it's addictive, whether or not you know him.
YP Rating: 4/5