Yes, Taverna Giulia is still there (Vicolo dell'Oro 23) and still renowned...but keep in mind that they serve Genovese and not Roman cooking:)
When eating in Rome, try to keep away from tourist traps, which are very diffused and, sadly, much more expensive than in other Italian cities. A good traditional cooking can be found at:
-Checchino dal 1887 (Via Monte Testaccio 30)
-Papà Giovanni (Via dei Sediari 4)
-Sora Lella (Via Ponte Quattro Capi 16)
-Sabatini (Piazza Santa Maria in Trastevere 18)
-Piperno (Via Monte de'Cenci 9), Roman-Hebrew cooking
If you're looking for "elegant" restaurants, the top is "La Pergola" at the Hilton Hotel (Via Cadiolo 101), otherwise:
-Alberto Ciarla (Piazza San Cosimato 40), fish
-Coriolano (Via Ancona 14)
-Quinzi e Gabrieli (Via delle Coppelle 6), fish
-Rosetta (Via della Rosetta 8), fish
A curiosity? There is in Rome a restaurant runned by nuns...not that typical as they serve a French-ethnic cooking, but anyway it can be fun:
-Eau Vive (Via Monterone 85)
As for Tuscany, you'll be located between Siena and Arezzo since Montebenichi (are you lodging at the Castelletto di Montebenichi?) is about 30 km far from both.
If you visit Siena and have some time left (I don't think so considering how amazing it is, but who knows?) forget the restaurants and give a look to the Enoteca Italiana, which is one of the greatest Enoteche in Italy.
Tuscany has some of the top restaurants in Italy, so if you want to eat great those are the best:
-Enoteca Pinchiorri in Firenze (Via Ghibellina 87), outstanding for elegance, food and WINES! Of course very expensive...
-Gambero Rosso in San Vincenzo, to tell the truth it's not just round the corner for you since it's on the seaside between Livorno and Grosseto, but another wonderful place;
-The same for Caino in Montemerano (Grosseto)
-Another very good restaurant in Firenze is "Cibreo", Via dei Macci 118.
Like Roma, also Firenze is obviously full of tourist traps, but since the Tuscan cooking is usually better than the Roman, maybe you could enjoy also some famous "typical" restaurants even if they have worsened a lot during the last decades due to the hordes of tourists;)
The most famous of those places is "Latini", via Palchetti 6r.
Three good restaurants, close to your location, are:
-Locanda Arnolfo in Colle Val D'Elsa (25 km from Siena)
-Antonio in Castelnuovo Berardenga (20 km from Siena), fish
-Locanda dell'Amorosa (Località Svolta L'Amorosa, Sinalunga) between Siena and Arezzo
And for traditional Tuscan cooking:
-Torrino (Località il Torrino, Arezzo)
-Albergaccio di Castellina (Castellina in Chianti, near Siena)
-Osteria di Rendola (Località Rendola, Montevarchi, 30 km from Arezzo)
I'd like to tell you more about those places, but unluckily I haven't time enough :(
Anyway, hope this helps...and have a good time!