Offal isn't the novelty it was in Boston a few years ago; you'll find it in all sort of places now. A few ideas:
Toro and Coppa, both in the South End, are under the same exec chef, do a nice variety of organ meats in myriad configurations.
Craigie on Main near Central Square is a highly-regarded indie that is very snout to tail. Speaking of tail, the pig's tails and whole head here are delicious.
Estragon, like the nearby Toro, also does Spanish-style offal dishes, like pringa, pig's head and delicious versions of tongue, fatback, cod cheeks, and chicken livers.
Staff Meal is a food truck that regularly features variety meats in its dishes; my favorite of our local mobile operators.
KO Prime near Park Street. sweetbreads done like chicken nuggets.
Tripe in a Florentine stew at Trattoria Toscana and many other North End Italians. Had this recently: lovely.
Buffalo chicken livers have popped up on a few menus; I like them at Green Street in Central Square.
Beef tongue in sauce at El Oriental de Cuba in Jamaica Plain.
Gastropubs like The Gallows (South End) and Russell House Tavern (Harvard Square) routinely feature organ meats in one form or another.
Tacos of beef tongue or chicharrones at many Salvadoran taquerias; these are easiest to find in East Boston and East Somerville.
Pork skin (bi) banh mi at Pho Viet in the Allston Super 88 food court. Pate and head cheese banh mi at many places.
Beef cheek tacos especialies at Taqueria El Amigo in Waltham, my favorite dish there.
Various animal bits on the charcuterie and salumi plates at The Butcher Shop, Taberna de Haro, Bin 26, and many others.
Mondongo (Puerto Rican tripe soup) and stewed oxtail at Izzy's near Kendall Square.
Pan-sauteed chicken livers at the Franklin Cafe in the South End and South Boston.
Chopped chicken liver at Rubin's Kosher Deli in Brookline.
Tripe and sweetbreads and occasionally brains at many of the traditional French places: Petit Robert, Pierrot, Troquet, Gaslight, etc.
Stewed cow's foot and goat's head soup at Suya Joint, a Nigerian place in Roslindale. (I didn't get to try either of these yet.)
Chicken hearts done rodizio style at Brazilian charrascarias like Mix Flavor Grill in Somerville.
Menudo, a common weekend special at many Mexican places (favored as a hangover remedy), notably Taqueria Jalisco in East Boston.
Intestine dishes in many Taiwanese and Korean restaurants.
The amazing frankfurters at Speed's Famous Hot Dog Wagon in Newmarket Square.
Chinatown -- too many to mention, though the Cantonese and Taiwanese joint reliably serve parts.
Gamjatang (Korean pork spine stew) at Hanmaru in Allston.
Cocido (offal stew) at El Embajador, a Dominican place in Jamaica Plain.
Hope that helps!
re: MC Slim JB
MC Slim JB as far as I know Mix Flavor Grill has been closed for almost a year.
The quintessial Brazilian offal dish would be Feijoada. If I buy it out its usually at Gaucho in East Somerville or another buffet, although Cafe Brazil in Allston serves it properly. Chicken Hearts certainly come next in popularity and Midwest Grill has a inexpensive bar plate with just charcoal grilled hearts and fried yuca, but "caldo de mocoto" a thick broth made with cow's hooves is a very well known bar speciality -- Cafe Tropical in Metrowest is a good option for that. White beans and tripe, dobradinha is common (not certain a favorite here, but have liked Brazil on Ferry in the past and I might try Cafe Tropical too, seems decent at Terras Brasilis). Least common of all locally, but a great find that I haven't tried -- Nab discovered that a Brazilian butcher (acougue) in Quincy is making morcela branca (sausage made with cooked pork offal). Beef liver is common on bar and appetizer menus, but I prefer Portugese liver (more often made with calves liver too).
The Portuguese side offers feijoada a transmontana which is made with red beans instead of the Brazilian black and appropriate chouricos, plus they also offer dobradinha also slightly different from the Brazilian version. J+J's in Somerville is a reasonable place for both and they often serve a side of gizzards as an opening.
In Salvadoran and other central american restauants around you can get find Sopa de Pata de Res which similar to Caribbean versions of mondongo which have hooves, it often has tripe.
BTW, in most of Latin American when you buy a stewing chicken (and often a more tender chicken) you get the feet (as well as the head and other bits). So properly most broths should have that mouthfeel and even the feet in there, but around here you'll have to search for somewhere offering "sopa de gallina india" on the weekends or something equivalent and ask for the feet. I actually like the feet stewed with yuca, but that is more of a home preparation.
Update: teezeetoo's post reminded me that Tango in Arlington has decent sweetbreads, although they sometimes overcook them a touch (I am not current on my ethnic, much less fine dining in Boston, so posting more to the ethnic although I have enjoyed offal in the past year at Toro, Coppa, Estragon, tongue at Rendezvous and more in the past year.)
great lists but my favorites would be Foie Gras torchon at Clink, pigshead toast at Strip T's, sweetbreads at La Voile, and roasted bone marrow and the offal of the day at ESK. Craigie does a lot of clever things with pig parts and offal and, as MC says, you'll find it in some form on most menus.
re: MC Slim JB
Recently, my favourite dish, offal or not, was "kat-a-kat" over at Darbar. It's lamb livers, kidneys, heart and ground meat cooked in a variety of spices on a blazing wok-like pan. Also at Darbar are "maghaz masala", or soft-scrambled brains ina lightly tomato-based curry, and "paya" which are goat or beef trotters in a gelatinous spicy gravy.
Another favourite of mine is the "laap khua" at Thai North - a cooked laap, pig innards (liver, heart, etc) stir-fried in a sensational spice mix.
re: MC Slim JB
Craigie on Main is having their annual "Road Less Travelled" dinner tonight --- last year's version included lots and lots and lots of offal.
Anticuchos (grilled Peruvian beef hearts) are excellent at Rincón Limeño in East Boston, and one of the better dishes on the otherwise generally mediocre menu at Machu Picchu in Somerville.
Taqueria Jalisco also has a number of offal fillings for tacos available every day, including beef head.
Abbondanza in Everett makes a terrific Roman-style (beef) tripe stew.
Many Chinese restaurants have good offal --- Red Peppers in Framingham has excellent beef larynx as well as more typical Sichuan offal offerings such as the fu qi fei pian (beef tripe and tendon in a spicy sauce).
There's lots of excellent Korean offal too, often in soups, but Myong Dong 1st Ave in Brighton also has tasty salty, fried chicken giblets.
At Shangri La in Belmont: (Brunch/Dim Sum) Pig Ear Salad in Hot Sauce, Pig Tripe Salad in Hot Sauce, Cattle Tendon in Hot Sauce, Shredded Beef Tripe in Hot Sauce, Steamed Spiced Chitterlings, Fried Chitterlings, Taiwanese Chitterling Vermicelli Soup; (Dinner) Sauteed Sour Mustard Tripe Soup, Crispy Intestine, Intestine and Pork Blood with Mustard in Hot Pot ,,,