My new wife and I will be honeymooning in Rome this April. We are looking for recommendations for a quiet hotel in the $250-$300 price range. Thank you.

Hotel Via S. Anselmo is lovely.
Very romantic - beautiful clawfooted bathtubs surrounded by tulle canopies.
Lovely little outside patio where you can have drinks in the evening. Beautiful outdoor garden for breakfast with their own lemon and orange trees.
Short walk to the Pyramide subway station. You can walk to the Colliseum. Actually we walked everywhere, downtown and to the Vatican.

I'll second the san anselmo (or either of their other two properties also on the Aventine)

You wake up to birdsong instead of traffic. You're surrounded by trees and attractive villas. You're atop the Aventine, with cool breezes but plenty of sunlight.

Off the beaten path but still close enough to get to any sights in short enough time. Lovely park atop the hill, the San Anselmo monastery, the Knights of Malta "keyhole" Sta. Sabina ... and easy access to Volpetti in Testaccio (the famous food store), as well as good restaurants in Trastevere.

Don't know how high the price has climbed, but check them out:
http://www.aventinohotels.com/

Their rates are well within your range.

The san Anselmo has been redone recently (was always attractive) and is now a bit over the top for my taste. I actually like the lower priced Hotel Aventino's simpler decoration better.

Our one time in Rome was also in April, and I chose the del Senato. We overlooked the Pantheon - perfect! Great largely pedestrian area, near lots of restaurants, double-glazed windows so we never heard any night-time noise, and we walked to all the sights.