If you're heading into Adams Morgan from Calvert Street Bridge, there are perfectly fine bike lanes on Calvert Street. But the walkways through Walter Pierce Park* make for a decent protected cycletrack as well. Call it the scenic route.
(*Or as my daughter calls it, 'Shady Park', since the tree canopy in the playground area makes it great on hot days.)
I should also mention, since it's hard to tell, this pair looked trés chic.