Blue dye, milk paint, satin paint and high gloss paint are all going to give you different interpretations of "blue."

You need to decide what look and feel you want.

Personally, I'd order them unfinished. Whether you are going to paint or dye is going to further determine wood species choice.

There are blue wood dyes. I use them. Get them at a woodworking store called Woodcraft.

On line also.

However, the type of wood has a HUGE effect on the final coloe. Oak and maple look totally different.

Then there is finishing. Oil based finishes will cause an amber tint, dulling the blue. Water based finishes dry clear, as a rule.

Blue dye will often look very similar to milk paint if the wood is lightly sanded after being dyed.

You know, I saw an exhibition of "Blue Guitars" at the Smithsonian years ago. Luthiers were invited to make a blue-color instrument. It proved to be one of the more difficult finishes to get perfect. The wood's natural color (they all started with a very light-toned wood like maple, spruce, etc.) still came through and many had a green tone.
Little-known fact: Kraftmaid, for one, offers an unfinished option, so you can decorate any way you like. Maybe others offer this "delete option".
Casey

Thanks for the advice, I'll look into and se if unfinished is an option. I hadn't really thought about the kind of look and feel I want - not green or amber so I guess I will stay away from oil.
Thanks again for the help, time for more research (wasting time on the computer)

Yes, most water-borne acrylics tend to have a blue cast to them. Some have an added amber, so stay away from them. Also look for one that is KCMA certified, meaning it passes a test for Kitchen Cabinet Manufacturer's Association.

I would get unfinished cabinets then order the blue analine powder, the one you mix with water. You can order from Woodcraft like Handymac explained. Then you need to get a large piece of the same kind of wood or plywood your cabinet is made of. Mix the dye and and try different strengths. You may also get some other colors like black or red to tweek your color. If you want to mix two colors, be sure to add water to make a base for each color. Don't mix two powders together. Make individual colors first then mix the liquids. It's kinda fun and you can learn usefull information.

Painting them will be a much easier option than trying to stain or dye them. Paint is opaque however, so if you do want to see the grain, the stain or dye is the better option. It's not nearly as DIY friendly though.