Dolomites to Trieste - Tricky path! Any suggestions?

There is a good reason for that. There are not any railroad tracks for the trains to run on. You don't say where you want to start this Dolomite visit or what you see. I see three routes.
1) Take train from Trieste to Conegliano, change to train to Ponte Alpi-Polpet, change train to Calazo-Pieve di Cadore, then bus to Cortina.
2) Take the train to Venice Mestre where you get a train to Bassano, then change to a train to Trento through the mountains.
3) Take the local train to Udine, then trains to Villach, Spittal, Lienz, then San Candido in the northern Dolomites These two are both round about ways.

What?! trains require tracks to run on them? :)
Yes, I perceived that to be the issue, and I assume it is a rugged and mountainous terrain.

We are flexible in our Dolomite terminus - we'll be staying in Ortisei, but I have an interest in adding Trento to the path, or perhaps alternatively Dobbiaco, for different reasons.

What about:
Cortina -> Calalzo -> Conegliano -> Udine ->Trieste

Seems like it would work. The question is would I WANT to do this, and would any of those places (Conegliano, Udine) be worthy stopovers? Curious adventure or back-woods nightmare?

Just edited. I initially thought you were going the other way. Then its a matter of getting around the Dolomites by bus. Haven't been to Udine. Trento is worth a stop.

AHA - trento to Bassano is also a good idea... thanks! :)

Bigger question is: What is worth seeing in the Eastern Dolomites? Anything?

Belluno and Maniago and Udine look interesting but none of the guide books are selling me on a visit.

So you are from New York City, one of the most congested cities in North America.

So why wouldn't you rent a car to visit the Dolomites, which would be the most sensible thing to do?

Is it because of fear of the intense traffic across the Bassano bridge during rush hour?

Or have you heard of Chris Christie planning to move to Italy to do some traffic studies on bridges of the Veneto region?

I think renting a car would greatly enhance your Dolomites' experience.

Precisely, Roberto. Because we are from NYC, we don't own or drive a car at all, so any driving anywhere feels unusual... I have never rented a car abroad, and it's quite out of our comfort zone. However, I may be persuaded to rent one for the period we are in the Dolomites - there are a number of things we want to do that are simply dependent upon it.

Here is a big fat file with the transport map and schedules for buses and trains in Alto Adige-Suedtirol.
https://www.sii.bz.it/sites/default/files/fahrplaene_orari_2015_web_0.pdf
Its in Italian and German, but a bus schedule is a bus schedule. Notes might need a little google translate work. Page 12 & 13 show the schematic transport map. This is until June 20, when the summer schedule, not out yet, takes over.
Here is site for lift schedule in Val Gardena. The summer schedule is not out yet, but you can see 2014.
http://www.valgardena.it/en/hiking-biking/walking-trekking/lifts/

Renting a car abroad is like renting it in America. The only difference is the shift, which is manual on most European cars as Europeans don't like to drive automatic cars (but you can specifically request an automatic transmission car, which tend to come on larger models only). Driving on the Dolomites is not difficult. Traffic is not heavy and average speeds are low due to the presence of curves.

Thank you, Sam! I speak German and can read italian alright too. Very helpful timetable.

Roberto, good heads-up -- I indeed do NOT drive shift ;)

I can't give you any on-the-ground advice, but I have been researching a potential visit to this region in October, and have found some driving tour itineraries in the Lonely Planet Italy's Best Trips and Backroads Italy books covering the road from Tarviso in the north to Trieste in the south. My research suggests that although it's possible to rely on public transportation in this region, it's much better to have a car, particularly if you want to get off the beaten path.