What is the most effective product to remove dog urine odors?  There
are so many products available and I would like to know others'
experience based on scientific reasons for success.

Scientific reason? Sorry, can't help you there, but I can tell you
from years of experience with a number of different pets my kids have
had growing up that Woolite Pet Stain Carpet & Upholstery Cleaner
works GREAT. It does the job and I have no complaints!

http://www.drugstore.com/qxp89409_333181_sespider/woolite/pet_stain_carpet_and_upholstery_cleaner.htm

tutuzdad-ga

I would say from experience to use Acetic Acid(vinegar) diluted with a
little bit of water. First blot the spot until it is completely dry.
After that pour a little bit of vinegar on the spot, blot that until
it is completely dry then rinse with water and blot that until it is
completely dry. hope this helps!

atlanta113: Try 3% hydrogen peroxide. It is inexpensive, harmless to
fabrics, leaves no residue, and has many uses.
monroe22

imo, simple green for the stain, and ozium spray for the odor.

Replacing carpet with vinyl, laminate flooring, or tile is a certain
fix.  Most of the cost is the installation charge.  Learning to do it
yourself will pay for the required tools.  Knowledge of flooring is a
very convenient skill for the pet owner.  Good luck.

I would highly recommend a product named "Sox" manufactured by Hagen. 
It doesn't just mask odors, but contains an enzyme that disolves the
odor causing agent... or so they claim.  Either way, it's been very
effective with removing the smell of cat pee, and snake excrements.

Natures Miracle Stain and Odor Remover.

After getting your old dog urine odor issues eliminated from surfaces, 
a product that has been most helpful in our household is Odor Free, 
which uses Mojave Yucca's natural odor-reduction properties to prevent
future dog & cat odors from forming (gas, body, urine, fecal & bad breath).

Good luck with the old odors you're trying to remove!  Your local
pet store might have some good ideas.