Solid Hardwood, Engineered and Laminate Flooring - Hardwood on concrete slab

I have a concrete slab that has a carpeted dining room and a kitchen with sheet vinyl. I would like to install 3/4" hardwood in the kitchen over the vinyl.

What is the recommended way to do this?

A major concern is the difference in elevation between the carpet and the hardwood and the resulting tripping hazard.

First of all, is this slab on grade, above grade, or below grade? If it's below, forget 3/4" solid wood.
Assuming it's on or above grade, you will need to install a new subfloor over the slab to nail the wood to. Some adhesive manufacturers will say you can glue solid strip flooring, but I would never advise it...it's a failure waiting to happen.
Personally, if I was going to do solid on this job, I would leave the vinyl in place as a vapor barrier, lay 5/8" plywood sheeting in one direction, lay another layer of 5/8" perpindicular to the first layer and nail those together. Now you have a suitable substrate.

You're going to raise the floor 2" total doing that, though. Personally, I would look for an engineered product with a sawn veneer....that way you get a lower profile, a more stable product, and the look of the solid floor you want.

The way it is generally done here on at grade slabs is to glue 2X4s flat to the slab on 16" centers and nail the hardwood flooring directly to the 2X4s. It seems to work fine without a sub floor but others I'm sure wood disagree but that is how almost all wood on slab floors are done here. Never actually seen it done any other way..

How the 2X4s are traditionally glued is with tar or cut back asphaltic adhesive poured on the whole floor and the 2 X4s set into it. Awfully messy. I just use heavy duty construction adhesive on the backs of the 2X4s.