Since Amazon changed their Terms Of Service around incentivized reviews, many sellers have been very nervous about asking for reviews at all. This is understandable given how valuable some Amazon businesses are, and how they are totally reliant on Amazon.
In my own Amazon business, I've been asking for reviews from customers for years. I've changed nothing since the update because in my view, at least at this point, there is no issue with asking for reviews - so long as you do not request positive reviews or offer any benefit for doing so.
Here's a short video of Jon (ZonGuru CEO) and I talking about this change and how we think it will impact the market:
Ultimately, we believe that this change is in the best interests of the customer, and it is upon that premise that Amazon has grown to become the most powerful online retailer in the world. Long term, this change is a great investment in the business, a business we as sellers are intrinsically linked to.
All of that being said, asking for an Amazon review still has to be handled carefully. Consumers, particularly in the US, are tired of being sold to over and over again. If you're going to email your customers through Seller Central after they buy from you, be mindful of the tone you adopt.
I personally use ZonGuru to send customer service emails to my Amazon customers, but my tone is always super-low-key and the messages are short. By doing this, I sound like a small business owner who really does care that their product arrived OK and that I want to know if there are any issues. In this way I am generating hundreds of organic reviews every year and I'm building a great brand on Amazon.