Not that familiar with Frightened Rabbit. Love every The National record I have other than the first one. Have gotten more airplay from me than any other artist over the last year. Here are some albums I have purchased in the last couple of months that I like:
Givers--In Light
My Morning Jacket--Circuital
REM--Collapse Into Now
Bon Iver
Fleet Foxes
Ben Harper--Give Til It's Gone
Yesterday, bought Rave On Buddy Holly mp3 for $3.99 on Amazon. Still at that price. Covers by The Black Keys, Paul McCartney, Julian Casablancas, My Morning Jacket, Modest Mouse, Lou Reed, Fiona Apple, Florence & the Machine, Cee Lo, She & Him, etc. Very good.
Just listened to the Yuck album on Spinner. Like it a lot. Need to listen again when I can crank it up. Has a Dinosaur Jr. feel.
I know none of these bands sound like The National, but somebody needed to reply to you.
Also, check out Townie's top 50 songs so far in 2011 and his Mixtape series. Good stuff on both.
Something that probably only I would like: Eddie Vedder-Ukulele Songs