And the caveat to this is that many "accelerated" schools will train you to pass a checkride, and just a week later you may have lost the skills and ability to pass it again, and in many cases the checkride wasn't really a fair test of your abilities and training, just a hobbled-together-"you didn't kill me, so you pass" type situation. Unfortunately, I've found this to be common at various schools, rather than really ensuring that the applicant knows his stuff and can do it well. Except for a few very gifted individuals that have to be VERY careful about how they utilize their skills and knowledge, most people take a while to develop the correct skills for the appropriate certificate and rating. Learning "fast" or "cheap" may hurt you down the line when you do a sim check-out and can't fly partial panel on "steam guages" or correctly interpret directions, clearances and charts.

But, the overall message is true, they don't care where you got your training, as long as you perform to the standard and have the required hours.