In my last house I repainted painted Ikea cabinets. I used Kilz primer so I didn't have to sand them. They were in my office but held up pretty well for the 10 years I was in the house. I assembled first and then painted them, but I just read a hack online where the poster said to paint them first. You still might be limited in your sizes, but you wouldn't have to upgrade the drawers.

Barker cabinets has good reviews here and moderate-low prices. CliqStudios (although components app. not made in USA) have good reviews but I don't think come unpainted. Conestoga has also gotten good mentions here, can be bought from various vendors (I got quote from Cabinet Authority) and I think but again not sure can be purchased unfinished-- also in the moderate-low range, about like Barker..
Did you google Kitchen Cabinet Depot BBB? that will bring up any Better Business Bureau complaints or ratings.

I'm using Home Depot unfinished cabinets in a low budget laundry room redo. We just finished painting last night and they are getting hung tomorrow, so I will try to post pics next week when the counters go in. It was a little challenge getting the sizes to work, so if you don't need custom sizes, they can do the job. I also used these in my master bathroom- 2 separate 30 inch sink bases. They are standard kitchen depth, so a little deeper than most vanities, but they work for us! I painted them black and topped them with carrera marble remnants. I love the look and they have held up fine- 4 years now!

If you are moderately handy ... check ana-white.com for base cabinet plans.

You can alter the sizes.

http://ana-white.com/2012/01/plans/36quot-corner-base-easy-reach-kitchen-cabinet-basic-model

Here is a link that might be useful: 36

If Ikea has the price range that you need, it seems that it is possible to piece together different elements to create a size closer to what you need. For example, I figured out how to get my 21" pantry out of a 15" pantry case mixed with 21" base cabs and drawers, and it still would have been a significant cost savings over Barker.

They do have 30" and 36" base drawer units BTW, they just have a split top drawer.

I feel like you can get Conestoga unfinished. Worth a call/email. Brian Long at the Cabinet Joint helped (and continues to help) us. These have lovely interiors, too. You get the hang of assembling them fairly quickly (first couple are a little scary, try without glue first time). Quick lead times, too. brian@thecabinetjoint.com

Conestoga RTA:

Good luck!

One more thought: we were able to customize drawer widths and heights at reasonable up charge.

Add Scherr's to the list (many threads here).

Another good option, possibly less expensive and completely customizable, is CabParts. They were featured in an article in Fine Homebuilding, used for an architect's own DIY kitchen.

I very nearly got these, but ended up getting Barker cabinets with unfinished doors, which we then had a local craftsman stain and finish. The Barker website made ordering far easier (than CabParts) which left me feeling more confident that I'd ordered what I needed. The price was very close to Conestoga, a bit more than CabParts.

Here is a link that might be useful: CabParts website

Thanks for the continued suggestions. I've browsed them all, and, on lazygardens' suggestion, have been looking more into building my own bases. I'm completely confident on uppers, and on lower boxes, just inexperienced on drawers. Multiple youtube videos later, on multiple different drawer construction methods, it's seeming feasible. Or there's Drawer Depot, that sells kits, any size, for $30 per drawer. We're only talking 3 drawer bases, a trash pullout and a sink front.

I'm also thinking of just doing a site-built stretch of cabinets instead of individual boxes. I'll keep sketching and googling...