disclaimer: I am a developer, with personal interest for UI and UX. So my main competencies are on the dev side: feel free to comment and let me know if something I say is questionable.
What kind of style you want your UI to have?
Using rounded corners VS square corners have IMO more implication than just solving the "blend with background" issue, I'm just listing a couple of them, but probably if you step a bit backwards from "what style should I have for this interface" to "what do I communicate using this interface" you will be able to make a choice that is simply coherent with more general questions.
Application style and Brand
Your app will be inserted in a wider emotional context. Maybe you already have (or you will need to define) a brand, a tone and a communication style that should be consistent acorss the app and the entire brand (if you have a web counterpart, or something like that)
Rounded corners have a friendlier, smoother, more childish tone. This can obviously be tuned and mitigated, but still something round communicates a softer tone.
Square corners are elegant, simpler, bold.
Again, depending on the tone you're seeking and the emotions and feeling you want to convey, one can be better than the other.
the new trend on android following the L design style seems to go towards a bolder approach, in which the crisp borders fit better. mobile design and tech enthusiast may be happy to see similar style in new apps, so if your target is this niche it may be interesting to explore more this kind of approach, and use maybe shadows / cards to be in line with the platform.
On the other side, rounded corners are a bit more web 2.0-ish, that is now a not-so-new style, but has spread out and had been able to spread wide also in paper-based communication etc.
So at the moment rounded corners may better for other niches or people that are not early adopters, since it is a style almost everyone has been somehow already exposed to. (I had recently been asked explicitly by a major carrier for rounded corners, that I personally find a bit old-looking. It's not clear to me if this is because the requirements were coming from someone that was not forward looking, or because they evaluated that rounded corners were better fitting their users, but still, it has some value)