I'm not an expert or anything, and I'm not familiar with those guys you mentioned, but the first thing that comes to mind is the "tambourine in there somewhere." My drummer puts a tambo on top of his hi-hat sometimes.
I think you're talking about a basic, stand-up rockabilly drum setup, no? Anyway, the basic recording technique is the same for any kind of kit. You put two mics on overhead boom stands, equidistant from the center of the kick, maybe a foot or so to either side of the kit, and about 6 or 7 feet high. Record these on separate tracks, and later, mix them with a spread, anywhere from 30/30 out, depending on what other instrumentation you have, and the sound you're after. The spread gives a sense of movement, or stereo.
Mic the snare and kick, and bring up these tracks in the mix to round out the drum sound. You can mic the top and bottom of the snare, and blend the two tracks for the sound you want. If you're just using one mic on the snare, mic the top. You'll get some bleed from the other drums in these mics, but it'll be mostly the drum you mic. Mic the kick from the front, not the back where the pedal is. My drummer's kick has a hole in the front head, about 6", I put the mic a few inches in front of that.
What kind of mics to use could fill up many pages of another thread, but use what you have, even stage mics can give a decent sound. Experiment with placement of the mics, it can make a big difference, as can the acoustics of the room you're in. In the mix, you'll probably want to EQ a lot of it, especially the kick. Keep the levels out of the red! You can always bring them up in the mix, but it's impossible to undo oversaturation.
Of course, there are other ways of doing it, but this is a common technique, and I could go on a lot more about finer points. The are lots of how-to vids on Youtube.
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