When you travel (or call) somewhere new, instead of consulting a timezone map to set your watch or know if it's safe to call someone, you'll consult a guide to local "time equivalents" and set your expectations based on that. "Let's see, it's 08:00UTC now and my time guide says locals at that location usually each lunch around 17:00UTC, so if I call Bob there now, it's the middle of the night for him"
The use of "tomorrow" and "yesterday" will be confusing in parts of the world where at 24/00 o'clock it isn't night. Convincing people will be difficult.
But for just the the EU it would work nicely.
I could maybe argue "well, 'yesterday' and 'tomorrow' aren't very precise anyway - you need the entire context of the conversation to understand what they're referring to", but they're well established and understood... Darn.
It really seems the simplest way to go. But I guess because of history it's hard for people to conceptualize how it would work. (Outside of HNers that is)