When I'm near the Colosseum, I eat at the unpretentious La Taverna Dei Quaranta
up the Caelian Hill at Via Claudia 24. I've always found the menu interesting and the food reasonably priced and delicious.

From my trip report in 2007:
Above the Colosseum is Hostaria de Nerone (a Frommers recommendation on via delle Terme di Tito). This was a good choice for a quick lunch after the Forum tour. I had the spaghetti all'Amatriciana which was quite good and veal saltimbocca alla romana which I wrote in my journal "was to die for". So I guess it was good. I didn't write down how much this was, but I recall it being quite reasonable (in line with my other choices). A nice location too if you've just left the Colosseum and are moving on to St. Pietro en Vincoli.

We enjoyed Naumachia www.naumachiaroma.com

We love the Taverna dei Fori Imperiali
http://dinersjournal.blogs.nytimes.com/2006/10/16/rome-picks-and-pans-part-two/

Thanks very much for these suggestions. And more questions, if you're willing:
Do these restaurants take reservations?
If it's very hot when we're in Rome, which of these might tend to be most cool?
Do you happen to know which might be open on Sunday?
Also: Does anyone know a restaurant called La Piazetta in the same vicinity?
Thank you!

Taverna dei Fori Imperiali takes reservations and you should make them. It's air-conditioned and open on Sunday.
http://www.latavernadeiforiimperiali.com/inglese/index.html

Thanks very much for these suggestions. And more questions, if you're willing:
Do these restaurants take reservations?
If it's very hot when we're in Rome, which of these might tend to be most cool?
Do you happen to know which might be open on Sunday?
Also: Does anyone know a restaurant called La Piazetta in the same vicinity?
Thank you!

Topping

"By the way, I hope that Italian dinner hour isn't as late as in Spanish? "

After 8 is the most popular and we always make reservations as we prefer to have dinner between 8 and 9, very busy times. Many restaurants do not even open until 7pm unless they are specifically geared to tourists.

Not sure if things have changed since Nov. of 2008 but we seldom dined outside as that is where the heavy smokers are and for us, that is a deal breaker.

This is an incredible restaurant in the Jewish Ghetto which is an easy walk from the Colosseum: http://www.giggetto.it All I can say is....delish.....

I forgot to add that you can walk through the ruins in the Jewish Ghetto to get to Giggetto and if you eat outside, you're eating right at the edge of them.

"Many restaurants do not even open until 7pm unless they are specifically geared to tourists."
Some restaurants do not even open until 8.30 pm.

Another vote for the Taverna dei 40. Nice food.