This is very possible; you define the URI scheme in your AndroidManifest.xml, using the <data> element. You setup an intent filter with the <data> element filled out, and you'll be able to create your own scheme. (More on intent filters and intent resolution here.)
Here's a short example:
<activity android:name=".MyUriActivity">
<intent-filter>
<action android:name="android.intent.action.VIEW" />
<category android:name="android.intent.category.DEFAULT" />
<category android:name="android.intent.category.BROWSABLE" />
<data android:scheme="myapp" android:host="path" />
</intent-filter>
</activity>
As per how implicit intents work, you need to define at least one action and one category as well; here I picked VIEW as the action (though it could be anything), and made sure to add the DEFAULT category (as this is required for all implicit intents). Also notice how I added the category BROWSABLE - this is not necessary, but it will allow your URIs to be openable from the browser (a nifty feature).