I've been mentally putting together insights about this experience (one in particular prompting this post) and wanted to compile them. One of the most helpful tips that I read was that when the baby fusses, assume first that he needs to pee rather than assuming he needs to nurse. In reality, a baby pees far more frequently than he nurses, so it's a better bet that that's the problem. So we always try to pee the baby before I nurse him. It was also helpful to realize that Dmitri fusses BEFORE he actually pees, not because he has a wet diaper (as we had previously assumed was the case with babies fussing because they "need to be changed"). When he's asleep and starts stirring or squirming, it's actually often because he has to pee. In the past, I would have nursed him back to sleep without thinking about him needing to pee (fitting with our general idea of "Nursing is the solution even when hunger is not the problem"). But my new philosophy is to check to see if he needs to pee. I have been very pleasantly surprised with the results of this experiment. I have many times now found that if I pop him on the pot when he fusses or squirms while sleeping, he'll pee and go straight back to sleep (no nursing necessary).
The other very reliable sign that he needs to eliminate is when he stops nursing. Often, he'll be nursing (when hungry, not just to calm down or for whatever other reason) and he'll just stop, popping off the breast. This almost always means he needs to go (often to poop, actually). I assume that this is at least partially due to the gastro-colic reflex. Poops are, I think, easier to catch because the signs are so much more obvious (grunting, farting) and because we make the practice of always trying to have him "finish" a poop on the toilet even if he's started in his diaper. We actually don't have to deal with poopy diapers very often anymore.
By far the most helpful thing we've done, though, is leaving Dmitri without a diaper cover. I have noted for a while now that leaving him without a cover greatly increases the percentage of pees that I catch (from catching a minority of pees--maybe 30%--to a majority--sometimes as many as 75-80%). We still catch pees when he wears a cover, but the diaper is almost always wet when we go to catch a pee. (Any catch is fun, but the most fun is when you catch a pee and get to put back on the same dry diaper.) I've been trying to figure out why this seems to make such a difference. My first suspicion was that I'm simply more vigilant when Dmitri isn't wearing a cover because I don't want to get peed on. Also, I think it's helpful to know right away when he's peed because it's useful in terms of timing (if he's just peed, I know he probably won't go again for at least 20 minutes if not an hour). I know it's also helpful to have the get-the-diaper-off, get-him-on-the-pot process be quicker, and forgoing the cover helps with that (as does having a pot right where I'm sitting, rather than having to walk him to the bathroom). But I think that there's something on his end, too, that makes having no cover helpful. Observing his sleeping has led me to this conclusion. I have now tried several times to leave him without a cover for naps (and, more recently, at night as well). I figured that the coverless diaper (MotherEase one-size's, by far our favourites for a variety of reasons) would keep a fountain from occurring. I also laid him down on a square of thick wool blanket that is inside a pillowcase. This serves as a moisture barrier between him and the bed, just in case (because I hate changing sheets). I was surprised when, several times, after sleeping for 2-3 hours, I would find him dry. When he fussed as he was waking, I'd pee him and just put his dry diaper back on. Sometimes he would go right back to sleep. Sometimes the nap was over (but 2-3 hours is long enough for a nap, I think). Last night, I tried this at night and found it to work similarly. He slept from 10pm to 3am, completely dry. I actually woke him up to pee and nurse at three even though he wasn't stirring. I have almost never found him to have a dry diaper after that many hours of sleep when he wears a cover, only when he's coverless. Which makes me think that he somehow experiences it differently.
Another small insight (this is maybe too obvious, but I'll say it anyway): he doesn't pee when crying. If we're trying to pee him when he's all upset, it's crucial to calm him down first. Peeing requires relaxation.
Also, to address a common misconception: we are not doing this to get him to be potty trained super early. We just don't want him to get used to eliminating in his clothing in the first place, and we want to reduce the amount of time that he spends sitting in a wet diaper (given how often infants pee, I've realized that they sit in wet diapers the majority of the time. This must be why so many people want diapers that "feel dry" to the kid).
Feel free to ask questions in the comments.