I'm playing through all the H3 campaigns (already finshed H1 and H2), and I'm really bummed out that certain campaigns don't allow you to carry your heroes to the next scenario. To me, the most entertaining aspect of H3 is customizing a hero with an eye on the end game.

Currently playing the Liberation campaign, on the third scenario with my third new hero. I'm having a LOT less fun with this campaign because of it.

Does anyone have a list of the campaigns in which you can carry heroes over from scenario to scenario?

I don't recall any of the Restoration of Erathia campaigns allowing you to carry heroes between scenarios. The Armageddon's Blade campaigns do carry heroes between scenarios (although it's a little weird with the actual Armageddon's Blade campaign since that one covers several storylines, so although heroes are carried through you're often jumping between them). The Shadow of Death campaigns also carry heroes through scenarios within a campaign, but although the same heroes are then used again in later campaigns the game doesn't carry through the heroes from the earlier campaigns (which can leave you with high-level heroes with some crappy skill selections). Also, the Heroes Chronicles (standalone HoMM3 expansion/campaign pack) campaigns carry through the hero within each campaign.

Also, both HoMM4 and HoMM5 carry through heroes in every campaign.

Actually, most of them do. The only ones that don't are "Liberation", "Long Live the King", and "Seeds of Discontent".

Granted, "Spoils of War" and "Dungeons and Devils" are a bit quirky, because you play the first two scenarios of each campaign in either order and then carry over the heroes from only one of them to the third scenario.

"Long Live the Queen" and "Song for the Father" each have you playing the scenarios in a specific order, with the heroes carrying over between scenarios, so in that regard they most closely resemble the AB and SoD campaigns.

Guess it's been a while since I've played the RoE campaigns. Thanks for the correction.

Simple advise: DO NOT PLAY RoE campaigns in Heroes III Complete. These campaigns are broken in Heroes III Complete.
Solution:
1. Install original Heroes III RoE v1.0 from original CD (released Feb. 16 1999).
2. Play these campaigns and feel the difference. Enjoy the high quality videos only available in RoE campaigns!

I believe GOG.com fixed that problem with the latest patch. Videos are now high quality for me and they do not reset after I exit the game.