Other than reading MathSciNet regularly, is there a way to get notified when one of my papers gets a review on MathSciNet?
While you could do as @Geoff Robinson suggests and look at MathSciNet regularly, in early 2017, we plan to have email alerts enabled on MathSciNet. The target is February 2017.
Other new features will be rolled out in January and February. The first demos will be at the Joint Mathematics Meetings in Atlanta.
Edward Dunne, Executive Editor, Mathematical Reviews
Though only semi-related to the question (couldn't submit it as a comment), I may take the liberty to add that for zbMATH subscribers there is an Atom Feed providing such regular updates for, e.g., subjects or authors. In your case, it would be sufficient to check your author ID at your profile https://zbmath.org/authors/?q=P%C3%A1lv%C3%B6lgyi+D%C3%B6m%C3%B6t%C3%B6r (i.e., Author ID palvolgyi.domotor) - and to use it to subscribe for https://zbmath.org/atom/ai/palvolgyi.domotor with you favourite news reader.
Olaf Teschke, Managing Editor, zbMATH