Nothing breaks my heart more than when I hear a GM complain that he wishes there were more roleplaying in his game. It's tragic because it's always so heartfelt and sincere and is almost always followed by said GM then introducing his new combat showpiece, hardcore dungeon crawl, or puppet show on rails. It hurts because the problem is so self-evident yet apparently completely unseen.
The solutions can also be painful, as the GM attempts to introduce "roleplaying encounters" into a game which neither wants nor needs them, but that attempt at a solution is emblematic of the problem. The idea that these other elements of gaming are somehow contradictory to roleplaying is pretty much entirely false. It's a case where there's plenty of correlation, but the cause is something else entirely.
Now, certainly there are some challenges - system mastery takes time and effort, and during the learning period, it's hard to focus on anything but the game. Sometimes a GM extends this period by following the path of the hard core - by constantly upping the challenge through increased mechanical complexity, he can extend the learning period indefinitely. That's a problem, yes, but not a problem with the games. Even the most complicated of games can reach mastery equilibrium in a reasonable timeframe with the right group or GM (or both).
But the real problem is the idea that the crunchy, fighty dungeon crawl is at odds with RP. It's nonsense, but it's deeply rooted nonsense that owes a lot to the history of the hobby and especially the history of published adventures. After all, books and movies can be full of high adventure and still support banter, character development, drama and so on - why is it a problem for games?
To understand the issue, let's take a moment to look at the heroes of fiction, especially adventure fiction. Generally speaking, they're presented with a challenge or challenges which they must overcome - not unlike adventurers. But the important part, often overlooked in gaming, is that part of the reason that the fiction is about these characters is because they are uniquely qualified to handle the challenge.
This idea of unique qualifications is a broad one because there are a lot of different things that make for UQ, and in fact in most fictions, the UQ is usually a result of a specific combination of non-unique qualifications. To illustrate that, consider that qualifications tend to fall into one of four loose categories - capability, knowledge, care, opportunity, and capability.
Capability is the first thing most gamers will think of. It means the hero is capable of tackling the problem either in the specific (he has the key to a specific lock) or in general (the problem is dangerous and he's badass). In gaming terms, we tend to jump right to thinking about this in terms of powers, skills and levels, but it can be much more nuanced.
Knowledge means that the hero sees the problem, often where others don't. Notably, it doesn't mean the hero knows _how_ to solve the problem - that's a form of capability - only that there's a problem to be solved.
Care means that the hero has a personal investment in the problem, a stake in the outcome which they're invested in. It might be because the problem affects them or those in their circle directly, or they might have a strong position on this particular type of problem. Care ends up being a kind of capability in certain types of fiction, especially noir detective stories - specifically, the protagonist has some moral backbone that allows them to pursue the problem rather than be consumed by the moral failings that surround him, like corruption.
Opportunity is, predictably, the opportunity to address the problem. It might be as simple as an issue of being in the right time and right place, but it might be part of a tangle of available time and conflicting responsibilities. Opportunity can muddle with capability very easily, especially when you start taking about authority or social position. A king can do a lot of things (capability and opportunity) but he may be bound by law (limit of capability) or unable to act due to other duties (lack of opportunity).
Look at any adventure fiction you like, and you'll find some combination of these in the protagonists. Sometimes you'll even find different combinations in different protagonists, and that can be pretty cool, but these unique qualifications provide implicit motivation and engagement for heroes in their own adventures.
Now, contrast this with the bog-standard dungeon crawl. At first blush, it looks like it demands several qualities - monsters must be fought (Capability), there's treasure to be gained (a kind of care) and the dungeon is conveniently nearby (opportunity) but they fall apart when you start looking for uniqueness.
See, by design, a published adventure needs to be able to be run through by any group of adventurers of a certain size and level, which means that, by design , it will demand no unique qualifications of adventurers (except perhaps those which it creates within its own bubble of fiction). Any other group of adventurers could do this (so much for capability), the reward is probably quite fungible (not much care left) and that leaves only opportunity. But thanks to the nature of geography and gaming, odds are good the dungeon of your level is going to "just happen" to be where you can get to it, so that feels like a fairly hollow oportunity.
But the problem is not dungeons! Not even super hard core crunchy ones. The problem is bad habits of framing. If you're a GM who wants to see more RP, then you need to start making the dungeons more engaging, and to do that, you need to figure out how to make the dungeon something that your specific group is uniquely qualified to address. Start from that foundation of generic threats and generic loot and start making it personal. Give your players a reason why _they_ are the ones going into this particular monster filled hole.
You'll find that RP emerges very naturally from that engagement, whatever system or style of play you use.