I love the Crowes, but I don't think there's a single signature tone I'd associate with them. It changed so much from album to album and tour to tour. Rich has played a lot of different amps over the years. Live he used Jubilees early on, then Matchless, Harry Joyce (Hiwatt clones), Vox, Reason, and Fender combos. I've seen footage of Marc Ford with various Marshall, Roccaforte, maybe Matchless and other amps. In the studio who knows, I would guess vintage Marshalls, Fenders, Vox and Matchless were on the first three or four albums. If you're covering more recent stuff like Warpaint or Before the Frost, I believe Luther Dickinson was playing an ODS style amp like a Fuchs (or at least he was using them live).
Unless you're going for something really specific, I think I'd start with either one of the AC flavored amps, a JTM45, or a 59 Bassman matched with an appropriate cab. Set it clean or just breaking up and hit it with pedals for heavier stuff and leads (maybe a TS or JanRay and a fuzz face). For effects: Rich always has a tape delay in his live rig, I'd also throw in a Clyde wah, comp, tremolo, and univibe. Rich also uses a kill-switch sort of effect and a volume pedal for swells on a few songs.