VoiceNote

Speak to capture a note! Voice note makes it very easy to speak to capture short memos and automatically send those notes to your favorite note organizing app. It is particularly useful for capturing a thought as quickly as possible, where you will have more time to sort it later.

VoiceNote:
Requires no internet connection, or any other permissions.
Can be launched from the lock screen or anywhere. It's not a widget, so it won't take any resources until it runs.
Can be configured to send notes directly to a few popular note apps.

You can select Evernote, Google Keep, Colornote, or just share the notes as text (so your device can let you pick the destination from any app that can receive it).

You need to have Android 4.1 or above because that's when Android put voice recognition directly on the device. That's pretty important, because relying on mobile data can make the voice recognition less reliable and the experience less fun.

Sure, there are other options to capture voice – Google Now has a “take a note” function. However, it fails when you are offline even when the voice recognition is directly on the device (anything with Android 4.1+). A lot of note apps have their own speak-to-text widget too, but those widgets are often bulky.

VoiceNote takes a minimalist approach which is not for everyone, but makes things extremely fast. You simply start it up, speak, and then the note gets sent off to its destination.

NOTE: VoiceNote has no main interface, other than the voice input dialog and the preferences screen. These keywords activate alternate functionality: help, preferences, settings, about.

VoiceNote is completely free (and ad-free) and is open source – get the source and build it yourself if you'd like: https://github.com/TimMackenzie/VoiceNote

If you have any suggestions for missing note organizing apps, please send a note!