I am building a cross-platform mobile app that interacts with a RESTful API, and I want to use OpenID Connect to authenticate my users. I will be building my own OpenID Connect
I think that the Hybrid flow from the OpenID Connect spec is probably the one which you want to use. OpenID Connect Core Spec.
This does rely upon having a configured return URI, but as James says you would use a custom URI scheme to enable the mobile OS to redirect after login to your own app. Your app would then have an access code which it can use to obtain access tokens as needed (assuming that you are using Oauth2 to protect your back-end API services which the mobile app uses).
There is a vulnerability which would allow a malicious app to hijack your URI scheme and grab the tokens, There is a draft spec to overcome that Proof Key for Code Exchange by OAuth Public Clients which is worth considering implementing.