My three year old isn't always in the mood to eat, and I work more or less full-time from home, so I don't have time to be a "coaxing waitress" (as I've told my husband). So that I'm not kept from the other myriad things I need to do, I have simplified. Here are some tips:
Make food that will more or less "keep" for a few (3) hours on the counter. Hard-boiled or scrambled eggs, cheese, veggie strips, cut-up fruit, crackers, simple sandwiches (nut butter and jelly), nuts, etc. No, they won't be as tasty as they might if the child had eaten them when they were first offered, but their palates aren't usually as easily offended as an adult's. And then, don't offer any substitutions for that meal.
Let them choose each choose one minor menu item each day, if it works for their age/level of maturity. Some kids will chose and then change their mind (mine doesn't get this sort of choices right now for that reason), so if that's the case, just skip this suggestion.
Offer a plate with at least two choices. Nothing backfires on us more than offering one choice, because we begin to second-guess ourselves if they are liking/off that food today, etc. Two or three choices on a plate is sufficient. (I've learned this one the hard way)
Let them help with cooking if they're interested. I didn't realize how helpful this would be, but my son's become more interested in trying new dishes and sampling the ingredients as we go along because he's working side-by-side with me. Kids can cut up some softer veggies and fruits with butter-knives, and love having their own cutting board/space. (I just reread your post and understand if this doesn't work for all meals--it's not always an inducement with mine either...perhaps have them make their own simple sandwiches?)
Put out the food and then, that's it. No offering any other alternatives. They will not starve themselves to death or die. They might be a bit cranky b/c they are hungry, and then you can redirect them back to the table. Leave the food at their place, too, unless you need the space and then just set it on the counter, explaining that you'll keep it for later.
By far and away, my two biggest tips for the families I've worked with: A. don't keep any food around the house that you don't want your kids to wheedle you for (if it's an adult treat, hide it well and eat it later when kids are in bed) and B. don't make an issue out of dessert. Many families I know do a 'backward dinner' where dessert is served as a small, 3 o'clock snack and then dinner later; the whole 'dragging our feet through a miserable dinner to get to dessert' usually lessens. My sis also has two 'dessert days' a week, where her boys can count on always having a dessert, regardless of meal consumption, etc. This has removed a large element of attempted negotiation on my nephews' part, as the conversation ends with "Today isn't a dessert day. We'll have dessert on Thursday."
Some kids need something else to do when they are eating. Offering some books/kid magazines to look at might help, or anything that you don't mind getting sticky fingers on. Busy wigglers might need to eat standing up at the table (mine can rarely sit down), but I would again leave it to them.
If you can, divorce yourself a bit from the emotional aspects of their eating. We feel like 'bad' parents when our kids don't eat, but they often are have phases where food just isn't important. Or they are so busy, they need quiet moments. If that's the case, see if they might like to have a story read to them. There are plenty of chapter books that are appropriate for 4&6s, and if you have a routine of reading 5 pages at lunch (not contingent on whether or not they are eating), this might change the setting just enough to get them settled into the meal. Or not. But if you aren't emotionally involved, it will put the onus back on them to settle this,because you aren't giving them anything to resist against. Thus, power struggle avoided. And then, get up from the table when you are done eating and move on with your day.
These suggestions won't work for every kid, but they've been lying around in the parenting toolbox, so I thought I'd share. Hope it goes well!