This image came from the designer who sold me my vanity and drew how to build vanity legs using trim pieces. These made the vanity 3 inches taller and up on three 'feet'. It sounds like you don't want to expose what is under your vanity so maybe you could come up with trim between each leg. Also to do this someone has to remove your vanity from your wall.

You can see the feet a little bit in this photo to see how it turned out.

We added 1/4" to the top of our vanity so that the top of the countertop would fit perfectly under the tile chair rail. Our little add-on was stained to match and is covered completely by the countertop overhang.

You could certainly do something similar, and with 3" to work with, I'd cover something structural (plain 2 x 4?) with a fabulous trim.

While the countertop is off, build a frame with 2x4s that you can set onto the existing frame and secure with screws (countersunk) down into the existing frame.

You will need a wider edging piece, but that would give you space for a strip of accent tiles or something.

It's the quick fix for tall folk :)

In our old kitchen we did much like what lazygardens describes on the island. (And by "we" I mean the guys who did the cabinets.) It was stock cabinets that they built a frame on, then added trim.

FYI: as you can see (I hope) from the corners of the island in the picture, if you use trim that is beveled or angled it increases the size of your counter to cover out to the edge of the trim. So keep this in mind when measuring for the counter and looking at trim.

As you probably figured out, your drawers will be a few inches below the counter. You can kind of see that in the second picture.

FWIW: when we did our reno and used custom cabinets, we raised the counters in the kitchen and all the bathrooms to 39". Definitely higher then standard (we're tall), especially for the bathrooms, but it's so much more comfortable for us.

Old kitchen island with trim to raise counter:
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