You're right, that would not be kosher. However, if you are self-published and have control over the Editorial Review section of the description then on Amazon you can add a snippet of the review in the product description and give credit to who/where it came from (if the person sending you the review agrees of course.) If it is already a publicly posted review on a blog then I don't think permission is necessary (fair use act and all that) but I don't think many bloggers would mind as long as you gave proper credit or you can ask them first if it doesn't feel right to you. You can do this at B&N too and they seem to have more leeway in this area. If you have a publisher than I don't know how much control you have over the description. The publisher may have to do this.
I know at least one author who did this with one of my reviews. I actually just stumbled upon it accidentally. PM me if you want a link to see what I mean.
To answer your original question... I don't think there is anything wrong with asking once, politely, and with no pressure. If they decline then leave it at that. You could also offer a little incentive like if they decide to leave a review you will send them a free ebook copy of your upcoming release. Nothing like a freebie to get people interested!
That reminds me... one of the authors on my goodreads friends list actually did something like this. She sent a note to everyone on her goodreads list offering a free ebook copy to anyone willing to do a review (via a smashwords coupon so they could download their format of choice). She seemed to get quite a few reviews from this method.
I have been an avid reader for 40 years and before I started blogging reviews, I did not know that you could review a book on Amazon or any of the other book sites. I would have loved to have had someone ask me, although I may have asked how it was done. One asking would have been amazing and I would have forever posted reviews after. Of course I love to write, but I do often think that readers are also writers.