1) Selecting relevant keywords and getting good reviews to move up app store search rankings. Also, having good app icon, copy and screenshots help get downloads. I got my app icon done at 99 designs. I am planning to add more custom graphics possibly with another 99 designs mobile app contest.
2) After a user uses the app 7 times you might prompt them to do a review. If they are using it that much, they will likely give a good review. My friend Evan that makes iExit said his rating improved by 1 and 1/2 stars.
3) Internet advertising using adwords, and unbounce campaigns can be used to A/B test for effective keywords, copy, and images. Also, search advertising is useful for harvesting demand, not creating it. So, may not be effective for a new type of app.
4) Viral features. Add a feature to "tell a friend email", unlock features if you post to social networks (eg. facebook connect), or include an ad for your app in an compelling feature that emails or posts to a social network.
5) Get app reviewed by mobile app blogs or niche blogs related to your product. You will need to prepare a media kit. There are services that will do this for you, although I don't know about the quality.
6) Include a link to email you complaints and feedback. If you implement the features let the people know and ask them to give you a good review.
7) Mobile Advertising like admob.
8) Make a free version of app or temporarily change the price to free for paid apps.
9) Make a video.
10) Get featured in the Apple app store. This, of course, is the holy grail, and may be difficult to achieve.
The main recurring advice is to make a great app, and that it may take several months of trying to get noticed. What strategies have you used?