I've never done one, but think lunch dates are just a bad idea. As you mentioned, no alcohol involved to help bring the guards down a bit, but it also sets a bad precedent that there is food now involved on all future meetings. If meet over lunch on a work day, also adds the stress of trying to rush conversation in that hour. Just don't see any real upside in midday first date lunches.
I see lunch dates differently.
1. I classify women 2 ways; daytime or evening material. If I really like them, or think they have potential, it's an evening date. If they are so so, it's a lunch date.
2. Leaving a lunch date is easy - you use the excuse of having to get back to work.
3. Single parents do not need to find a sitter for a lunch date.
4. Leaves your evenings open for a more desirable date or activity.
5. If your date works downtown, it's so convenient.
Of course there are disadvantages but each situation is different.