If there’s any doubt the New Orleans Suspects are the best kind of throwback, check the funky, jubilant version of the traditional hymn “Jesus on the Mainline” that closes their debut CD. The band pulls out everything to make the tune percolate, including a rollicking C.R. Gruver piano solo (which works in a Fess reference or two), drummer Willie Green’s second-lining and one of Reggie Scanlan’s typically supple basslines. Topping that are plenty of handclaps and call-and-response vocals, assuring us that Jesus will give you all you need if you just get him on the telephone. But then you realize that without a mainline phone, you may be out of luck: the hymn isn’t updated, so there’s no word on whether Jesus accepts emails or text messages nowadays.

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Buy the New Orleans Suspects on iTunes

Buy the New Orleans Suspects on Amazon

The Suspects are likely to be the last notable new band whose history goes back a certain distance, including members who’ve played with Professor Longhair and Earl King. Three New Orleans cornerstones are represented in the lineup: Green and Scanlan respectively hail from the Neville Brothers and the Radiators; guitarist Jack Eckert and saxophonist Kevin Harris (since replaced by Jeff Watkins) are both from the Dirty Dozen Brass Band. Keyboardist / singer Gruver is the only one without a big-league pedigree, but his vocals and solos prove to be one of the main assets. Given the history, this Suspects could easily have gone to the bank as a cover band — and indeed, they have been known to pull out a “Tipitina” or a “Hey Pocky Way” at shows.

But it’s clear from the CD that there’s some more serious musicology going on: each of the covers is identified by who recorded it first and by which version they’re referencing (in the case of “Money Honey,” it’s Ry Cooder’s version, one of many good options). The find of the batch is a 1972 Bobby Charles number, “All the Money” — Charles was likely writing about his then label head Albert Grossman (“He got all the money, and he won’t give me none”), but it’s as timely a sentiment as you could ask. Of the four band-written tunes, three are instrumentals and/or Mardi Gras grooves that find the band doing what comes naturally — making it sound easier than it is. But the exception, “36 Cars,” is also the standout: a classic-sounding rocker that has a nicely mysterious lyric, and echoes a number of bands — there’s some Dead, some Little Feat and certainly some Radiators — without getting too far into anyone else’s territory. It’s proof that familiar sounds and usual suspects can still be combined in exciting ways.